%%% -*-BibTeX-*-
%%% ====================================================================
%%% BibTeX-file{
%%% author = "Evgenija D. Popova",
%%% version = "0.6",
%%% date = "15 December 2014",
%%% filename = "pubs-WK6.bib",
%%% address = "Bulgarian Academy of Sciences,
%%% Institute of Mathematics and Informatics,
%%% Acad. G. Bonchev str., block 8,
%%% 1113 Sofia,
%%% Bulgaria,",
%%% telephone = "+359 889662835",
%%% FAX = "+359 2 9713649",
%%% URL = "http://www.math.bas.bg/~epopova",
%%% email = "epopova at math.bas.bg, epopova at bio.bas.bg,
%%% epopova at bas.bg",
%%% codetable = "ISO/ASCII",
%%% keywords = "BibTeX, bibliography, Walter Kraemer,
%%% Numerical Computation, Interval Computation,
%%% Enclosure Methods, Mathematical Software,
%%% Result Verification, Standard and Special Functions,
%%% Multiple/Arbitrary Precision Computations,
%%% High Performance Verified Computing,
%%% Computer Algebra Systems, C-XSC",
%%% license = "public domain",
%%% supported = "no",
%%% docstring = "This is a tentatively COMPLETE bibliography of
%%% Prof. Dr. Walter Kraemer from Bergisches Univ.
%%% Wuppertal, Germany.
%%%
%%% This bibliography was prepared by merging
%%% data from the TeX Users Group bibliography
%%% archive, the BibNet Project archive, the OCLC Contents1st
%%% database, the huge Karlsruhe computer science
%%% bibliography archive, the MathSciNet
%%% database, the journal Web pages and information
%%% provided by Dr. Werner Hofschuster from Bergisches Univ. Wuppertal.
%%% Some errors in the sources noted above
%%% have been corrected.
%%%
%%% Bibliography entries below for
%%% all articles that appeared in German use the
%%% original German titles.
%%%
%%% BibTeX citation tags are uniformly chosen as
%%% name:year:abbrev, where name is the family
%%% name of the first author or editor, year is a
%%% 4-digit number, and abbrev is a 3-letter
%%% condensation of important title words.
%%%
%%% This bibliography is sorted by year, and
%%% within each year, by author and title key.
%%%
%%%
%%% ====================================================================
%%% Abbreviations:
@String{today = "2014-12-15"}
@String{ack-edp = "Evgenija D. Popova
Bulgarian Academy of Sciences,
Institute of Mathematics and Informatics,
Acad. G. Bonchev str., block 8,
1113 Sofia,
Bulgaria,
Tel: +359 889662835 (mobile),
Fax: +359 2 9713649,
e-mail: \path|epopova@bio.bas.bg|,
\path|epopova@math.bas.bg|,
\path|epopova@bas.bg|,
URL: \path|http://www.math.bas.bg/~epopova/|"}
%%% ====================================================================
%%% Bibliography entries:
@TechReport{Bohlender:1981:PCS,
author = "G. Bohlender and K. Gr{\"u}ner and E. Kaucher and R.
Klatte and W. Kr{\"a}mer and U. Kulisch and W. L. Miranker
and S. M. Rump and Ch. Ullrich and {Wolff von Gudenberg}, J.",
title = {{Pascal-SC}: a {Pascal} for Contemporary Scientific Computation},
type = "Research Report",
number = "RC 9009",
institution = "IBM Thomas J. Watson Res. Cent.",
address = "Yorktown Heights, New York",
year = "1981",
bibdate = today,
bibsource = "http://liinwww.ira.uka.de/bibliography//Math/intarith.html;
http://liinwww.ira.uka.de/bibliography//Parallel/Eugene/index.html",
abstract = "Adds COMPLEX data type and operator overloading and exponentiation.",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)"
}
@TechReport{Bohlender:1982:MP,
author = "G. Bohlender and H. B{\"o}hm and K. Gr{\"u}ner and E.
Kaucher and R. Klatte and W. Kr{\"a}mer and U. Kulisch and
W. L. Miranker and S. M. Rump and Ch. Ullrich and {Wolff von Gudenberg}, J.",
title = {{Matrix Pascal}},
type = "Res. Rep.",
number = "RC 9577",
institution = "IBM Thomas J. Watson Res. Cent.",
address = "Yorktown Heights, New York",
year = "1982",
bibdate = today,
bibsource = "http://liinwww.ira.uka.de/bibliography//Math/intarith.html",
note = "Published in \cite{Bohlender:1983:MP}",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@TechReport{Bohlender:1982:PAS,
author = "G. Bohlender and H. B{\"o}hm and K. Gr{\"u}ner and E.
Kaucher and R. Klatte and W. Kr{\"a}mer and U. Kulisch and
S. M. Rump and Ch. Ullrich and {Wolff von Gudenberg}, J.",
title = {Proposal for Arithmetic Specification in {Fortran 8x}},
type = "{Interner Bericht Des Instituts F. Angew. Math.}",
institution = "Universit{\"a}t Karlsruhe",
address = "Karlsruhe, Germany",
year = "1982",
bibdate = today,
bibsource = "http://www.math.utah.edu/pub/tex/bib/fortran2.bib;
http://liinwww.ira.uka.de/bibliography//Math/intarith.html",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@TechReport{Bohlender:1983:AMS,
author = "G. Bohlender and H. B{\"o}hm and K. Braune and K. Gr{\"u}ner
and E. Kaucher and R. Klatte and W. Kr{\"a}mer and U.
Kulisch and W. L. Miranker and Ch. Ullrich and {Wolff von Gudenberg}, J.",
title = {Application Module: {S}cientific-{C}omputation for {Fortran}
8x. {M}odified Proposal for Arithmetic Specification
According to Guidelines of the {X3J3}-Meetings in {Tulsa and Chapel Hill}},
type = "Interner Bericht",
institution = "Instituts F. Angew. Math. Universit{\"a}t Karlsruhe",
address = "Karlsruhe, Germany",
year = "1983",
bibdate = today,
bibsource = "http://www.math.utah.edu/pub/tex/bib/fortran2.bib;
http://liinwww.ira.uka.de/bibliography//Math/intarith.html",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@incollection{Bohlender:1983:MP,
author = "G. Bohlender and H. B{\"o}hm and K. Gr{\"u}ner and E.
Kaucher and R. Klatte and W. Kr{\"a}mer and U. Kulisch and
W. L. Miranker and S. M. Rump and Ch. Ullrich and {Wolff von Gudenberg}, J.",
title = {{Matrix Pascal}},
booktitle = {A New Approach to Scientific Computation},
editor = "U. W. Kulisch and W. L. Miranker",
series = "Notes and Reports in Computer Science and Applied Mathematics",
volume = "7",
ISBN = "0124286607",
publisher = "Acad. Press , New York",
pages = "311--384",
year = "1983",
note = "see also \cite{Bohlender:1982:MP}",
remark = "Proceedings of the Symposium on a New Approach to Scientific Computation (Yorktown Heights, NY) 1982.08.03",
bibdate = today,
bibsource = "https://getinfo.de/app/subject-search?",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@InProceedings{Bohlender:1984:ASF,
author = "G. Bohlender and H. B{\"o}hm and K. Gr{\"u}ner and E.
Kaucher and R. Klatte and W. Kr{\"a}mer and U. Kulisch and
W. L. Miranker and S. M. Rump and Ch. Ullrich and {Wolff von Gudenberg}, J.",
editor = "B. Ford and J. C. Rault and F. Thomasset",
booktitle = "Tools, Methods and Languages For Scientific and Engineering Computation",
title = {Arithmetic Specification In {Fortran 8x}},
publisher = "Elsevier Sci. Publ., North Holland",
address = "Amsterdam, The Netherlands",
pages = "213--243",
year = "1984",
bibdate = today,
bibsource = "http://www.math.utah.edu/pub/tex/bib/fortran2.bib;
http://liinwww.ira.uka.de/bibliography//Math/intarith.html",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@TechReport{Kraemer:1984:DFA,
author = "Walter Kr{\"a}mer",
title = {{Dokumentation und Fehlerabsch{\"a}tzungen der hochgenauen
ACRITH (Intervall-) Standard\-funktionen (inverse Funktionen)}},
year = "1984",
type = "Booklet",
institution = "IBM Deutschland",
note = "220p.",
bibdate = today,
bibsource = "",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@InProceedings{Braune:1985:SFI,
author = "Klaus Braune and Walter Kr{\"a}mer",
title = {Standard Functions for Intervals with Maximum Accuracy},
booktitle = "Proc. 11th IMACS Congr. Syst. Simul. Sci. Comput.
Oslo, Norway, Aug. 5--9, 1985, Vol. I",
publisher = "",
address = "",
pages = "167--170",
year = "1985",
note = "see also \cite{Braune:1986:HAS}",
bibdate = today,
bibsource = "http://liinwww.ira.uka.de/bibliography//Math/intarith.html",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@incollection {Braune:1986:HAS,
AUTHOR = {Braune, Klaus and Kr{\"a}mer, Walter},
TITLE = {High-Accuracy Standard Functions for Intervals},
BOOKTITLE = {Computer Systems: Performance and Simulation},
EDITOR = {Ruschitzka, M.},
PAGES = {341--347},
PUBLISHER = {Elsevier Science Publishers B.V. (North-Holland)},
YEAR = {1986},
ISBN = {9780444700810},
NOTE = {see also \cite{Braune:1985:SFI}},
bibdate = today,
bibsource = "",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@TechReport{Kraemer:1986:FHK,
author = "Walter Kr{\"a}mer",
title = {{Fehlerabsch{\"a}tzungen der hochgenauen komplexen ACRITH
In\-ter\-vall\-stan\-dard\-funk\-tio\-nen (inverse Funktionen)}},
type = "Booklet",
PAGES = {1--220},
year = "1986",
institution = "IBM Deutschland",
bibdate = today,
bibsource = "",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@incollection {Braune:1987:HAS,
AUTHOR = {Braune, Klaus and Kr{\"a}mer, Walter},
TITLE = {High-accuracy standard functions for real and complex intervals},
BOOKTITLE = {Computerarithmetic: Scientific Computation and Programming Languages},
EDITOR = {Kaucher, E. and Kulisch, U. and Ullrich, Ch.},
PAGES = {81--114},
PUBLISHER = {Teubner, Stuttgart},
YEAR = {1987},
MRCLASS = {65G10},
bibdate = today,
bibsource = "http://www.ams.org/mathscinet/, MRNUMBER=904309 (88j:65094)",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@PhdThesis{Kraemer:1987:ISR,
author = "Walter Kr{\"a}mer",
title = {{Inverse Standardfunktionen f{\"u}r reelle und komplexe
Intervallargumente mit a priori Fehlerabsch{\"a}tzungen
f{\"u}r beliebige Datenformate}},
type = "Dissertation",
school = "Universit{\"a}t Karlsruhe",
address = "Karlsruhe, Germany",
pages = "",
year = "1987",
language = "German",
remark = "(Inverse Elementary Functions for Real and Complex Interval
Arguments with A-Priori Error Estimates for Arbitrary Data Formats)",
bibdate = today,
bibsource = "garbo.uwasa.fi:/pc/doc-soft/fpbiblio.txt;
http://www.math.utah.edu/pub/tex/bib/elefunt.bib",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@TechReport{Kraemer:1988:MRI,
author = "Walter Kr{\"a}mer",
title = {{M}ehrfachgenaue reelle und intervallm{\"a}{\ss}ige
staggered-correction {A}rithmetik mit zugeh{\"o}rigen {S}tandardfunktionen},
type = "Bericht",
year = "1988",
institution = "Inst. f. Angew. Mathematik, Universit{\"a}t Karlsruhe",
note = "pages 1--80",
bibdate = today,
bibsource = "http://liinwww.ira.uka.de/bibliography/Math/UIA.html",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@InProceedings{Kraemer:1988:ISF,
author = "Walter Kr{\"a}mer",
title = {Inverse standard functions for real and complex point
and interval arguments with dynamic accuracy},
editor = "Ulrich Kulisch and Hans J. Stetter",
booktitle = "Scientific Computation with Automatic Result Verification",
series = "Computing Supplementum",
publisher = "Springer, Vienna",
ISBN = "0-387-82063-9",
isbn-13 = "978-0-387-82063-7",
volume = "6",
pages = "185--212",
year = "1988",
CODEN = "COSPDM",
ISSN = "0344-8029",
DOI = "10.1007/978-3-7091-6957-5_16",
URL = "http://dx.doi.org/10.1007/978-3-7091-6957-5_16",
doi-URL = "http://dx.doi.org/10.1007/978-3-7091-6957-5_16",
remark = "Computer Arithmetic and Scientific Computation, 30 Sept.-2 Oct. 1987, Karlsruhe, West Germany,
sponsors: Karlsruhe Univ. and GAMM Committee.",
keywords = "Approximation errors; Dynamic accuracy; Error formula;
Interval arguments; Inverse standard functions; Monotonicity behaviour; Rounding errors",
abstract = "Algorithms to compute inverse standard functions to
arbitrary accuracy with safe error bounds are given.
Not only approximation errors but also all possible
rounding errors are considered. The desired accuracy of
the function as well as the base of the number system
used are parameters of the error formula. For
implementation it is only assumed that the four
elementary arithmetic operations are performed with a
certain number of correct digits of the function value.
The interval routines are constructed out of the point
routines considering the monotonicity behaviour of the
functions. The ambiguity of the complex functions is
briefly discussed.",
MRCLASS = "B0290B (Error analysis in numerical methods); B0290F
(Interpolation and function approximation); C4110
(Error analysis in numerical methods); C4130
(Interpolation and function approximation)",
thesaurus = "Error analysis; Function approximation",
bibdate = today,
bibsource = "http://www.math.utah.edu/pub/tex/bib/computing.bib;
http://liinwww.ira.uka.de/bibliography//Math/UIA.html;
http://liinwww.ira.uka.de/bibliography//Math/computer.arithmetic.1.html;
http://www.math.utah.edu/pub/tex/bib/elefunt.bib;
http://liinwww.ira.uka.de/bibliography//Math/intarith.html;
http://www.ams.org/mathscinet/, MRNUMBER = 1011721 (90m:65095)",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@incollection{Cordes:1989:VPE,
AUTHOR = {Cordes, D. and Kr{\"a}mer, Walter},
TITLE = {{Vom Problem zum Einschlie{\ss}ungsalgorithmus}},
BOOKTITLE = "{Wissenschaftliches Rechnen mit Ergebnisverifikation}",
EDITOR = {U. Kulisch},
PUBLISHER = {Vieweg, Braunschweig},
PAGES = {167--181},
YEAR = {1989},
language = {German},
bibdate = today,
bibsource = "",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@TechReport{Kraemer:1989:FEE,
author = "Walter Kr{\"a}mer and W. V. Walter",
title = {{FORTRAN-SC}: a {FORTRAN} Extension for
Engineering\slash Scientific Computation with Access to
{ACRITH}: General Information Notes and Sample Programs},
type = "Booklet",
institution = "IBM Deutschland GmbH",
address = "Stuttgart",
year = "1989",
note = "pp.~1--51",
bibdate = today,
bibsource = "http://www.math.utah.edu/pub/tex/bib/fortran2.bib",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@article{Kraemer:1989:FHA,
AUTHOR = {Kr{\"a}mer, Walter},
TITLE = {Fehlerschranken f{\"u}r h{\"a}ufig auftretende {A}pproximationsausdr{\"u}cke},
JOURNAL = {Z. Angew. Math. Mech.},
FJOURNAL = {Zeitschrift f{\"u}r Angewandte Mathematik und Mechanik (ZAMM)},
Publisher = {Wiley, Berlin},
VOLUME = {69},
YEAR = {1989},
NUMBER = {4},
PAGES = {t44--t47},
ISSN = {0044-2267; 0946-8463/e},
MRCLASS = {65D20; 33B15; 65G10},
language = {German},
bibdate = today,
bibsource = "https://zbmath.org/?q=an:0679.68063; https://zbmath.org/bibtex/04113987.bib",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@article{Kraemer:1990:BGP,
AUTHOR = {Kr{\"a}mer, Walter},
TITLE = {Berechnung der {G}ammafunktion {$\Gamma(x)$} f{\"u}r reelle {P}unkt- und {I}ntervallargumente},
JOURNAL = {Z. Angew. Math. Mech.},
FJOURNAL = {Zeitschrift f{\"u}r Angewandte Mathematik und Mechanik (ZAMM)},
Publisher = {Wiley, Berlin},
VOLUME = {70},
YEAR = {1990},
NUMBER = {6},
PAGES = {T581--T584},
ISSN = {0044-2267; 0946-8463/e},
MRCLASS = {65D20; 33B15; 65G10},
remark = {Bericht {\"u}ber die Wissenschaftliche Jahrestagung der GAMM (Karlsruhe, 1989)},
language = {German},
bibdate = today,
bibsource = "http://www.ams.org/mathscinet/, MRNUMBER=1070154 (91h:65034);
https://zbmath.org/?q=an:0728.65014; https://zbmath.org/bibtex/04203560.bib",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@incollection{Kraemer:1990:HAE,
AUTHOR = {Kr{\"a}mer, Walter},
TITLE = {Highly Accurate Evaluations of Program Parts with Applications},
BOOKTITLE = {Contributions to Computer Arithmetic and Self-Validating Numerical Methods},
SERIES = {IMACS Annals on Computing and Applied Mathematics},
Volume = {7},
EDITOR = {Ullrich, Ch.},
PAGES = {397--410},
PUBLISHER = {J.C. Baltzer AG, Basel},
YEAR = {1990},
remark = {Proceedings of the IMACS-GAMM-GI International Symposium
on Computer Arithmetic and Self-Validating Numerical Methods, Basel, Switzerland, 1989},
bibdate = today,
bibsource = "https://zbmath.org",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@article{Kraemer:1991:EPK,
AUTHOR = {Kr{\"a}mer, Walter},
TITLE = {Einschlu{\ss} eines {P}aares konjugiert komplexer {N}ullstellen eines reellen {P}olynoms},
JOURNAL = {Z. Angew. Math. Mech.},
FJOURNAL = {Zeitschrift f{\"u}r Angewandte Mathematik und Mechanik},
VOLUME = {71},
YEAR = {1991},
NUMBER = {6},
PAGES = {T820--T824},
ISSN = {0044-2267},
language = {German},
remark = {({E}nclosure of a pair of conjugate complex zeros of a real polynomial)},
keywords = {enclosure; conjugate complex zeros; real polynomial; Bairstow method; numerical example},
bibdate = today,
bibsource = "https://zbmath.org/?q=an:0735.65031; https://zbmath.org/bibtex/00027987.bib",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@article{Kraemer:1991:GBK,
AUTHOR = {Kr{\"a}mer, Walter},
TITLE = {{Genaue Berechnung von komplexen Polynomen in mehreren Variablen}},
JOURNAL = {Wissenschaftliche Zeitschrift der Technischen Hochschule Leipzig},
VOLUME = {Heft 9},
NUMBER = {},
YEAR = {1991},
PAGES = {401--407},
address = {Leipzig},
bibdate = today,
bibsource = "",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@incollection{Kraemer:1992:BSR,
AUTHOR = {Kr{\"a}mer, Walter},
TITLE = {Die {B}erechnung von {S}tandardfunktionen in {R}echenanlagen},
BOOKTITLE = {Jahrbuch {\"{U}}berblicke {M}athematik, 1992},
EDITOR = {Chatterji, S.D. and Kulisch, U. and Laugwitz, D. and Liedl, R. and Purkert, W.},
PAGES = {97--115},
PUBLISHER = {Vieweg, Braunschweig},
YEAR = {1992},
MRCLASS = {65D20; 65-04},
bibdate = today,
bibsource = "http://www.ams.org/mathscinet/, MRNUMBER=1156521",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@incollection{Kraemer:1992:CIB,
AUTHOR = {Kr{\"a}mer, Walter},
TITLE = {Computation of interval bounds for elliptic integrals},
BOOKTITLE = {Computer Arithmetic and Enclosure Methods ({O}ldenburg, 1991)},
EDITOR = {J. Herzberger and L. Atanassova},
PAGES = {289--298},
PUBLISHER = {North-Holland, Amsterdam},
YEAR = {1992},
MRCLASS = {65D20; 65G10},
remark = {Proceedings of the International Symposium on Computer Arithmetic
and Scientific Computation, Oldenburg 1991 (SCAN91)},
bibdate = today,
bibsource = "http://www.ams.org/mathscinet/, MRNUMBER=1207828",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@incollection{Kraemer:1992:EPS,
AUTHOR = {Kr{\"a}mer, Walter},
TITLE = {Evaluation of polynomials in several variables with high accuracy},
BOOKTITLE = {Computer Arithmetic, Scientific Computation and Mathematical Modelling},
SERIES = {IMACS Ann. Comput. Appl. Math.},
EDITOR = {Kaucher, E. and Markov, S.M. and Mayer, G.},
VOLUME = {12},
PAGES = {239--249},
PUBLISHER = {Baltzer, Basel},
YEAR = {1991},
MRCLASS = {65D15; 65G10; 65Y99},
remark = {Proceedings of the IMACS-GAMM-IIASA International Conference
on Computer Arithmetic, Scientific Computation and Mathematical Modelling, 1990, Albena, Bulgaria},
bibdate = today,
bibsource = "http://liinwww.ira.uka.de/bibliography/Math/UIA.html;
http://www.ams.org/mathscinet/, MRNUMBER=1189145 (93h:65020)",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@incollection{Kraemer:1992:VSE,
AUTHOR = {Kr{\"a}mer, Walter},
TITLE = {Verified solution of eigenvalue problems with sparse matrices},
BOOKTITLE = {Computational and Applied Mathematics, I. Algorithms and Theory},
EDITOR = {Brezinski, C. and Kulisch, U.},
PUBLISHER = {North-Holland, Amsterdam},
ISBN = {0-444-89701-1/hbk},
PAGES = {277--287},
YEAR = {1992},
MRCLASS = {65F50; 65F10; 65G10},
remark = {Proceedings. Selected and revised papers from the 13th IMACS World Congress, Dublin, Ireland, July 1991.},
keywords = {numerical example; interval arithmetic; verified solution; eigenpair; sparse eigenproblem},
abstract = {A theorem of S. M. Rump on the enclosure of a simple eigenpair of a general matrix is modified so that sparse matrices can be treated efficiently. The exact solution of the sparse eigenproblem is enclosed by an interval vector with sharp bounds.},
bibdate = today,
bibsource = "https://zbmath.org/?q=an:0773.65023; https://zbmath.org/bibtex/00176411.bib;
http://www.ams.org/mathscinet/, MRNUMBER=1203352 (94c:65058)",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@TechReport{Kraemer:1993:BFK,
author = "Walter Kr{\"a}mer",
title = {{Die Berechnung von Funktionen und Konstanten in Rechenanlagen}},
year = "1993",
type = "Habilitationsschrift",
institution = "University of Karlsruhe",
address = "Karlsruhe, Germany",
language = "German",
remark = "93 S. Available in Karlsruher Institut f{\"u}r Technologie (KIT) - Bibliothek, Signatur: 95 DE 73",
bibdate = today,
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@incollection{Kraemer:1993:CIB,
AUTHOR = {Kr{\"a}mer, Walter and Barth, B.},
TITLE = {Computation of Interval Bounds for {W}eierstrass' Elliptic Function $\wp (z)$},
BOOKTITLE = {Validation Numerics: Theory and Applications},
EDITOR = {Albrecht, R. and Alefeld G. and Stetter, H.},
SERIES = {Computing. Supplementum},
VOLUME = {9},
PAGES = {147--159},
PUBLISHER = {Springer, Wien},
YEAR = {1993},
ISBN = "0-387-82451-0 (New York), 3-211-82451-0 (Vienna)",
isbn-13 = "978-0-387-82451-2 (New York), 978-3-211-82451-1 (Vienna)",
CODEN = "COSPDM",
ISSN = "0344-8029",
DOI = "10.1007/978-3-7091-6918-6_12",
URL = "http://dx.doi.org/10.1007/978-3-7091-6918-6_12",
doi-URL = "http://dx.doi.org/10.1007/978-3-7091-6918-6_12",
KEYWORDS = {Elliptic functions, enclosure methods, XSC-languages},
ABSTRACT = {A method to enclose the values of Weierstrass elliptic function $\wp (z) = \wp(z|g_2,g_3)$ for arbitrary complex invariants $g_2,g_3$ or arbitrary given zeros of the characteristic polynomial (arbitrary period lattices) is presented. The function is approximated by its truncated Laurent series at zero. An error bound is derived for the remainder term. If necessary, the periodicity of $\wp$, the homogeneity relations and the addition formulas are used to perform the reduction of the argument and the corresponding adaptation of the result.},
bibdate = today,
bibsource = "http://www.math.utah.edu/pub/tex/bib/computing.bib",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@ARTICLE{Kraemer:1993:EPL,
author = {Kr{\"a}mer, Walter},
title = {Eine portable {L}angzahl- und {L}angzahlintervallarithmetik mit {A}nwendungen},
journal = {Z. angew. Math. Mech.},
fjournal = {Zeitschrift f{\"u}r Angewandte Mathematik und Mechanik (ZAMM)},
publisher = {Wiley, Berlin},
year = {1993},
volume = {73},
number = {7--8},
pages = {T849--T853},
ISSN = {0044-2267; 0946-8463/e},
language = {German},
remark = {({A} portable multiple-precision arithmetic and multiple-precision interval arithmetic with applications)},
MRCLASS = {65G30, 65D20, 65Y15},
keywords = "multiple-precision interval arithmetic; PASCAL programs; elementary functions; result verification",
abstract = "",
bibdate = today,
bibsource = "https://zbmath.org/?q=an:0795.65024; https://zbmath.org/bibtex/00584521.bib",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@InCollection{Kraemer:1993:MPC,
author = {Kr{\"a}mer, Walter},
title = {Multiple-Precision Computations with Result Verification},
booktitle = {Scientific Computing with Automatic Result Verification},
series = {Mathematics in Science and Engineering},
editor = {Adams, E. and Kulisch, U.},
ISBN = {0-12-044210-8/hbk},
year = {1993},
volume = {189},
pages = {325--356},
publisher = {Academic Press Inc., Boston},
MRCLASS = {65D20; 65G30},
keywords = "number pi; Archimedes' method; rounding errors; interval arithmetic; arithmetic-geometrie mean; natural logarithm",
abstract = "",
bibdate = today,
bibsource = "http://liinwww.ira.uka.de/bibliography/Math/UIA.html;
https://zbmath.org/?q=an:0805.65027; https://zbmath.org/bibtex/00434854.bib",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@incollection{Kraemer:1993:NBS,
AUTHOR = {Kr{\"a}mer, Walter},
TITLE = {Numerische {B}erechnung von {S}chranken f{\"u}r $\pi$},
BOOKTITLE = {Jahrbuch {\"{U}}berblicke {M}athematik, 1993},
EDITOR = {Chatterji, S.D. and Fuchssteiner, B. and Kulisch, U. and Liedl, R.},
PAGES = {57--72},
PUBLISHER = {Vieweg, Braunschweig},
YEAR = {1993},
MRCLASS = {11Y60},
abstract = "",
bibdate = today,
bibsource = "http://www.ams.org/mathscinet/, MRNUMBER=1208107 (94e:11147)",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@TechReport{Bohlender:1994:GBA,
author = "G. Bohlender and Walter Kr{\"a}mer and W.~L.~Miranker",
title = {Grading of Basic Arithmetical Operations and Functions},
institution = "IBM Research Division, T.~J. Watson Research Center",
year = "1994",
number = "RC 19593 (86059)",
bibdate = today,
bibsource = "http://liinwww.ira.uka.de/bibliography/Math/computer.arithmetic.html",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@TechReport{Kraemer:1994:BBI,
author = "Walter Kr{\"a}mer",
title = {{Bericht {\"u}ber die Begutachtung des IWRMM im Dezember 1993}},
type = "Preprint",
number = "IWRMM 94/3",
year = "1994",
institution = "IWRMM, University of Karlsruhe",
address = "Karlsruhe, Germany",
language = "German",
bibdate = today,
bibsource = "",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@Book{Kraemer:1994:NTV,
author = "Walter Kr{\"a}mer and Kulisch, U. and Lohner, R.",
title = {Numerical Toolbox for Verified Computing II --- Advanced Numerical Problems},
year = "1994",
address = "Karlsruhe",
publisher = "unpublished draft",
URL = "http://www2.math.uni-wuppertal.de/wrswt/literatur/tb2.pdf",
bibdate = today,
bibsource = "",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@ARTICLE{Kraemer:1994:SWW,
author = {Kr{\"a}mer, Walter},
title = {{Schranken f{\"u}r Werte der Weierstra{\ss}schen $\wp$--Funktion}},
journal = {Z. angew. Math. Mech.},
fjournal = {ZAMM --- Journal of Applied Mathematics and Mechanics/Zeitschrift f{\"u}r Angewandte Mathematik und Mechanik},
publisher = {Wiley-VCH Verlag, Berlin},
year = {1994},
volume = {74},
number = {6},
pages = {t660--t662},
issn = {0044-2267; 0946-8463/e},
language = {German},
MRCLASS = {65D20, 65G30, 65E05, 30B50, 33E05},
keywords = "",
abstract = "",
bibdate = today,
bibsource = "https://zbmath.org/?q=an:0806.65011; https://zbmath.org/bibtex/00641610.bib",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@ARTICLE{Kraemer:1995:GVN,
author = {Kr{\"a}mer, Walter},
title = {Grundlagen des verifizierten numerischen {R}echnens ({B}ases of verified computing)},
journal = {Z. angew. Math. Mech.},
fjournal = {ZAMM --- Journal of Applied Mathematics and Mechanics/Zeitschrift f{\"u}r Angewandte Mathematik und Mechanik},
publisher = {Wiley-VCH Verlag},
year = {1995},
volume = {75},
number = {SUPPL. 2},
pages = {s425--s428},
issn = {1521-4001 (online)},
language = {German},
MRCLASS = {65G30, 65H05, 65D25},
keywords = "interval arithmetic; zeros of functions; automated differentiation",
abstract = "",
bibdate = today,
bibsource = "https://zbmath.org/?q=an:0850.65085; https://zbmath.org/bibtex/00799189.bib",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@ARTICLE{Kraemer:1995:VFE,
author = "Kr{\"a}mer, Walter",
title = {Validated function evaluation using polynomial approximation from truncated {C}hebyshev series},
journal = "Reliab. Comput.",
fjournal = "Reliable Computing",
year = "1995",
volume = "Supplementum",
pages = "113-113",
url = "http://www.cs.utep.edu/interval-comp/apic.95/kraemer.ps",
remark = "Proceedings of International Workshop on Applications of Interval Computations, El Paso, Texas, February 23-25, 1995",
bibdate = today,
bibsource = "",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@TechReport{Hofschuster:1996:RFA,
author = "Hofschuster, W. and Walter Kr{\"a}mer",
title = {{Ein rechnergest{\"u}tzter Fehlerkalk{\"u}l mit Anwendung auf ein genaues Tabellenverfahren}},
type = "Preprint",
number = "IWRMM 96/5",
year = "1996",
note = "35p.",
institution = "IWRMM, University of Karlsruhe",
address = "Karlsruhe, Germany",
language = "German",
bibdate = today,
bibsource = "",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@InProceedings{Kraemer:1996:CNI,
author = "Walter Kr{\"a}mer and Stefan Wedner",
title = {Computing Narrow Inclusions for {Cauchy} Principal Value Integrals},
editor = "G{\"o}tz Alefeld and Andreas Frommer and Bruno Lang",
booktitle = "Scientific Computing and Validated Numerics",
publisher = "Akademie Verlag, Berlin, Germany",
series = "Mathematical Research",
volume = "90",
pages = "45--51",
year = "1996",
ISBN = "3-05-501737-4",
isbn-13 = "978-3-05-501737-7",
ISSN = "0138-3019",
remark = "Proceedings of the International Symposium on
Scientific Computing, Computer Arithmetic and Validated
Numerics {SCAN}-95 held in Wuppertal, Germany, September 26--29, 1995",
MRCLASS = "65D30; 65G10",
abstract = "",
bibdate = today,
bibsource = "http://www.math.utah.edu/pub/tex/bib/fparith.bib;
OCLC Proceedings database;
http://www.ams.org/mathscinet/, MRNUMBER=1394224",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@TechReport{Kraemer:1996:SGA,
author = "Walter Kr{\"a}mer",
title = {{Sichere und genaue Absch{\"a}tzung des Approximationsfehlers bei rationalen Approximationen}},
type = "Bericht",
number = "3/1996",
institution = "Forschungsschwerpunkt Computerarithmetik, Intervallrechnung
und Numerische Algorithmen mit Ergebnisverifikation (CAVN)",
year = "1996",
address = "Karlsruhe, Germany",
language = "German",
bibdate = today,
bibsource = "",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@Article{Kraemer:1996:TAG,
title = {Two adaptive {G}auss-{L}egendre type algorithms for the verified computation of definite integrals},
author = "Walter Kr{\"a}mer and Stefan Wedner",
journal = "Reliab. Comput.",
fjournal = "Reliable Computing",
year = "1996",
number = "3",
volume = "2",
pages = "241--253",
ISSN = "1385-3139",
DOI = "10.1007/BF02391698",
URL = "http://dx,doi.org/10.1007/BF02391698",
doi-URL = "http://dx,doi.org/10.1007/BF02391698",
MRCLASS = "65D30; 65G10",
abstract = "We propose a two algorithms for computation of (sharp) enclosures of definite interevals: a local adaptive algorithm (LAA) and a global adaptive algorithm (GAA). Both algorithms are based on Gauss-Legendre quadrature. Error terms are bounded using automatic differentiation in combination with interval evaluations. Several numerical examples are presented; these examples include comparison with an adaptive interval Romberg scheme.",
bibdate = today,
bibsource = "http://dblp.uni-trier.de/db/journals/rc/rc2.html#KramerW96;
http://liinwww.ira.uka.de/bibliography//Misc/RC.html;
http://www.ams.org/mathscinet/, MRNUMBER=1403539 (97d:65008)",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@TechReport{Kraemer:1996:TWW,
author = "Walter Kr{\"a}mer and Niethammer, W. and {(editors)}",
title = {{Tagungsband zum Workshop Wissenschaftliches Rechnen
in den Ingenieurwissenschaften, Februar 1996, Karlsruhe}},
type = "Preprint",
number = "IWRMM 96/6",
year = "1996",
institution = "IWRMM, University of Karlsruhe",
address = "Karlsruhe, Germany",
language = "German",
bibdate = today,
bibsource = "",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@Book{Niethammer:1996:WRU,
author = "Niethammer, W. and Kr{\"a}mer, Walter and {(editors)}",
title = {{Wissenschaftliches Rechnen in der Universit{\"a}t Karlsruhe}},
year = "1996",
publisher = " Universit{\"a}t Karlsruhe",
bibdate = today,
bibsource = "",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@TechReport{Blomquist:1997:AGF,
author = "Blomquist, F. and Walter Kr{\"a}mer",
title = {{Algorithmen mit garantierten Fehlerschranken f{\"u}r die Fehler- und die komplement{\"a}re Fehlerfunktion}},
type = "Preprint",
number = "IWRMM 97/3",
year = "1997",
institution = "IWRMM, University of Karlsruhe",
address = "Karlsruhe, Germany",
language = "German",
bibdate = today,
bibsource = "",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@Article{Hofschuster:1997:COA,
title = {A Computer Oriented Approach to Get Sharp Reliable Error Bounds},
author = "Werner Hofschuster and Walter Kr{\"a}mer",
JOURNAL = "Reliab. Comput.",
fjournal = "Reliable Computing",
year = "1997",
number = "3",
volume = "3",
pages = "239--248",
DOI = "10.1023/A:1009966622475",
URL = "http://dx.doi.org/10.1023/A:1009966622475",
doi-URL = "http://dx.doi.org/10.1023/A:1009966622475",
ISSN = {1385-3139},
CODEN = {RCOMF8},
MRCLASS = {65G10},
remark = {Proceedings of the International Conference on Interval
Methods and Computer Aided Proofs in Science and
Engineering --- Interval'96 (W{\"u}rzburg, 1996)},
abstract = {We present interval methods to get reliable a priori error bounds for the machine evaluation of algorithms implementing some mathematical expression. The term expression not only means simple arithmetical expressions but also more complex program parts including loops or recursive structures (e.g. a complete elementary function routine). We sketch a method that can be used to get an upper bound for the approximation error of a polynomial or a rational approximation. We also discuss a method to compute worst case a priori error estimates for arbitrary IEEE double floating-point computations. Our theoretical results lead to reliable and easy to use public domain software tools. The application of these tools to an accurate table method shows that error bounds of high quality can be derived.},
bibdate = today,
bibsource = "http://liinwww.ira.uka.de/bibliography//Misc/RC.html;
http://dblp.uni-trier.de/db/journals/rc/rc3.html#HofschusterK97;
http://www.ams.org/mathscinet/, MRNUMBER=1616261 (99a:65062)",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@InProceedings{Hofschuster:1997:FPD,
author = "Werner Hofschuster and Walter Kr{\"a}mer",
title = {A Fast Public Domain Interval Library in {ANSI C}},
booktitle = "Proceedings of the 15th IMACS World Congress on
Scientific Computation, Modelling and Applied Mathematics",
editor = "A. Sydow",
year = "1997",
volume = "2",
pages = "395--400",
keywords = "",
abstract = "",
bibdate = today,
bibsource = "http://liinwww.ira.uka.de/bibliography/Math/auto.diff.html",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@InProceedings{Kraemer:1997:AWC,
author = "Walter Kr{\"a}mer",
title = {A Priori Worst-Case Error Bounds for Floating-Point Computations},
editor = "Tomas Lang and Jean-Michel Muller and Naofumi Takagi",
booktitle = "13th {IEEE} Symposium on Computer Arithmetic:
proceedings, July 6--9, 1997, Asilomar, California, {USA}",
volume = "13",
publisher = "IEEE Computer Society Press",
ISBN = "0-8186-7846-1, 0-8186-7847-X, 0-8186-7848-8",
isbn-13 = "978-0-8186-7846-2, 978-0-8186-7847-9, 978-0-8186-7848-6",
ISSN = "1063-6889",
series = "Symposium on Computer Arithmetic",
pages = "64--73",
year = "1997",
DOI = "10.1109/ARITH.1997.614880",
URL = "http://www.acsel-lab.com/arithmetic/arith13/papers/ARITH13_Kramer.pdf",
doi-URL = {http://dx.doi.org/10.1109/ARITH.1997.614880},
abstract = {A new technique for the a-priori calculation of rigorous error bounds for floating-point computations is introduced. The theorems given in this paper, combined with interval arithmetic, lead to the implementation of reliable software routines which enable the user to compute the desired error bounds automatically by a suitable computer program. As a prominent example, a table-lookup algorithm for calculating the function $exp(x)-1$ that has been published by Tang (1992) is analyzed using these new tools. The result shows the high quality of the new approach},
keywords = {error analysis; floating point arithmetic; function evaluation; software reliability; subroutines; table lookup; a-priori worst-case error bounds; exponential function; floating-point computations; interval arithmetic; reliable software routines; table-lookup algorithm; algorithm design and analysis; computer errors; digital arithmetic; equations; error analysis; floating-point arithmetic; linear systems; reliability theory},
note = "See also \cite{Kraemer:1998:AWC}.",
bibdate = today,
bibsource = "http://dblp.uni-trier.de/db/conf/arith/arith1997.html#Kramer97;
http://www.math.utah.edu/pub/tex/bib/fparith.bib; OCLC Proceedings database",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@TechReport{Kraemer:1997:EFZ,
author = "Walter Kr{\"a}mer",
title = {{Eine Fehlerfaktorarithmetik f{\"u}r zuverl{\"a}ssige a priori Fehlerabsch{\"a}tzungen}},
type = "Bericht",
number = "5/1997",
institution = "Forschungsschwerpunkt Computerarithmetik, Intervallrechnung
und Numerische Algorithmen mit Ergebnisverifikation (CAVN)",
year = "1997",
address = "Karlsruhe, Germany",
language = "German",
bibdate = today,
bibsource = "",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@InProceedings{Kraemer:1997:ERC,
author = "Walter Kr{\"a}mer",
title = {Effective range computations for rational functions},
booktitle = "Proceedings of the 15th IMACS World Congress on
Scientific Computation, Modelling and Applied Mathematics",
editor = "A. Sydow",
year = "1997",
volume = "2",
pages = "421--426",
keywords = "",
abstract = "",
bibdate = today,
bibsource = "",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@TechReport{Bantle:1998:KVA,
author = "Bantle, A. and Walter Kr{\"a}mer",
title = {{Ein Kalk{\"u}l f{\"u}r verl{\"a}{\ss}liche absolute und relative Fehler\-absch{\"a}tzungen}},
type = "Preprint",
number = "IWRMM 98/5",
year = "1998",
institution = "IWRMM, University of Karlsruhe",
address = "Karlsruhe, Germany",
language = "German",
bibdate = today,
bibsource = "",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@TechReport{Hofschuster:1998:FSP,
author = "Hofschuster, Werner and Walter Kr{\"a}mer",
title = {{{\tt FI\_LIB}, eine schnelle und portable Funktionsbibliothek f{\"u}r reelle Argumente und
reelle Intervalle im IEEE-double-Format}},
type = "Preprint",
number = "IWRMM 98/7",
year = "1998",
note = "227p.",
institution = "IWRMM, University of Karlsruhe",
address = "Karlsruhe, Germany",
language = "German",
bibdate = today,
bibsource = "",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@ARTICLE{Kraemer:1998:AAF,
author = {Kr{\"a}mer, Walter},
title = {Automatisierte a-priori-{F}ehlerabsch{\"a}tzung bei gleitkommam{\"a}{\ss}iger {A}usdrucksauswertung},
journal = {Z. angew. Math. Mech.},
fjournal = {ZAMM --- Journal of Applied Mathematics and Mechanics / Zeitschrift fur Angewandte Mathematik und Mechanik},
publisher = {Wiley-VCH Verlag},
year = {1998},
volume = {78},
number = {SUPPL. 3},
pages = {979--980},
issn = {1521-4001 (online)},
DOI = {10.1002/zamm.19980781559},
doi-url = {http://dx.doi.org/10.1002/zamm.19980781559},
url = {http://onlinelibrary.wiley.com/doi/10.1002/zamm.19980781559/pdf},
remark = {Supplement: Short Communications in Applied Mathematics, GAMM'97, University of Regensburg, Germany March 24-27, 1997; article first published online: 23 DEC 2010},
language = {German},
keywords = "",
abstract = "",
bibdate = today,
bibsource = "http://onlinelibrary.wiley.com/documentcitationdownload?publicationDoi=10.1002/%28ISSN%291521-4001&doi=10.1002/zamm.19980781559&type=journal",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@Article{Kraemer:1998:AWC,
author = "Kr{\"a}mer, Walter",
title = {A priori worst case error bounds for floating-point computations},
JOURNAL = {IEEE Trans. Comput.},
fjournal = "IEEE Transactions on Computers",
volume = "47",
number = "7",
pages = "750--756",
month = jul,
year = "1998",
CODEN = "ITCOB4",
ISSN = "0018-9340 (print), 1557-9956 (electronic)",
issn-l = "0018-9340",
DOI = "10.1109/12.709374",
URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=709374",
doi-url = "http://dx.doi.org/10.1109/12.709374",
journal-url = "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12",
MRCLASS = {65Y10; 65G10; 68M07},
keywords = {automation of error analysis; elementary function algorithms; rigorous error bounds; table-lookup algorithms},
abstract = "A new technique for the a priori calculation of
rigorous error bounds for floating-point computations
is introduced. The theorems given in the paper combined
with interval arithmetic lead to the implementation of
reliable software routines, which enable the user to
compute the desired error bounds automatically by a
suitable computer program. As a prominent example, a
table-lookup algorithm for calculating the function
$exp(x) - 1$ that has been published by P. T. P. Tang
(1992) is analyzed using these new tools. The result
shows the high quality of the new approach",
note = "see also \cite{Kraemer:1997:AWC}",
bibdate = today,
bibsource = "http://ieeexplore.ieee.org/;
http://www.computer.org/tc/;
http://www.math.utah.edu/pub/tex/bib/ieeetranscomput1990.bib;
http://www.math.utah.edu/pub/tex/bib/toms.bib;
http://dblp.uni-trier.de/db/journals/tc/tc47.html#Kramer98;
http://www.ams.org/mathscinet/, MRNUMBER=1635507 (99f:65210)",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@Article{Kraemer:1998:CEA,
author = "Walter Kr{\"a}mer",
title = {Constructive Error Analysis},
journal = "J.UCS",
fjournal = "J.UCS: Journal of Universal Computer Science",
volume = "4",
number = "2",
pages = "147--163",
day = "28",
month = feb,
year = "1998",
CODEN = "????",
ISSN = "0948-695X (print), 0948-6968 (electronic)",
issn-l = "0948-6968",
DOI = "10.3217/jucs-004-02-0147",
URL = "http://www.jucs.org/jucs_4_2/constructive_error_analysis/Kraemer_W.pdf",
doi-URL = "http://dx.doi.org/10.3217/jucs-004-02-0147",
other-url = "http://medoc.springer.de:8000/jucs/jucs_4_2/constructive_error_analysis",
MRCLASS = {65G40},
keywords = {floating point computations; rigorous error bounds; scientific computing; validated numerics},
abstract = {Rigorous a priori error bounds for floating-point computations are derived. We will show that using interval tools in combination with function and operator over-loading such bounds can be computed on a computer automatically in a very convenient way. The bounds are of worst case type. They hold uniformly for the specified domain of input values. That means, whenever the floating point computation is repeated later on with any set of point input values from that domain the difference of the exact result and the computed result is guaranteed to be smaller than the a priori error bound. Our techniques can be used to get reliable a priori error bounds for already existing program code. Here, loops, recursion, and iterations are allowed. To demonstrate the power of the methods several examples are given.},
remark = {{SCAN-97} (Lyon)},
bibdate = today,
bibsource = "http://www.math.utah.edu/pub/tex/bib/jucs.bib;
http://dblp.uni-trier.de/db/journals/jucs/jucs4.html#Kramer97;
http://www.ams.org/mathscinet/, MRNUMBER=1661844 (2000d:65082)",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@TechReport{Bantle:1999:IHB,
author = "Bantle, A. and Kr{\"a}mer, Walter",
title = {Implementierung, {H}andhabung und {B}eispielanwendungen eines verl{\"a}sslichen {V}orw{\"a}rtsfehlerkalk{\"u}ls},
type = "Bericht",
number = "2/1999",
institution = "Forschungsschwerpunkt Computerarithmetik, Intervallrechnung
und Numerische Algorithmen mit Ergebnisverifikation",
address = "Karlsruhe, Germany",
year = "1999",
language = "German",
keywords = "",
abstract = "",
bibdate = today,
bibsource = "",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@TechReport{Brauer:1999:RAB,
author = "Br{\"a}uer, M. and Kr{\"a}mer, Walter",
title = {R{\"u}ckw{\"a}rtsmethode zur automatischen {B}erechnung von worst-case {F}ehlerschranken},
type = "Bericht",
number = "3/1999",
institution = "Forschungsschwerpunkt Computerarithmetik, Intervallrechnung
und Numerische Algorithmen mit Ergebnisverifikation (CAVN)",
address = "Karlsruhe, Germany",
year = "1999",
language = "German",
keywords = "",
abstract = "",
bibdate = today,
bibsource = "",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@inproceedings{Budding:1999:VLA,
title = {Verifikation von {L}{\"o}sungen ausgew{\"a}hlter {P}robleme aus der {M}odellierung von {M}anipulatoren},
author = {B{\"u}dding, G. and Fausten, D. and Kr{\"a}mer, W. and M{\"o}llers, Th. and Traczinski, H.},
booktitle = {Schriftenreihe des Fachbereichs Mathematik, Universit{\"a}t Duisburg, Gesamthochschule; 463},
pages = {},
year = {1999},
remark = {18. Symposium: Modellierung und Simulation von Verkehr : Bad Honnef am 18. und 19. November 1999.},
keywords = "",
abstract = "",
bibdate = today,
bibsource = "",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@TechReport{Geulig:1999:IME,
author = "I. Geulig and Walter Kr{\"a}mer",
title = {{Intervallrechnung in Maple --- Die Erweiterung
intpakX zum Paket intpak der Share-Library}},
type = "Report",
institution = "IWRMM, University of Karlsruhe",
address = "Karlsruhe, Germany",
year = "1999",
language = "German",
remark = "({Interval} arithmetic in {Maple} --- The extension
{{\tt intpakX}} to the Share Library package {\tt intpak})",
bibdate = today,
bibsource = "http://www.math.utah.edu/pub/tex/bib/maple-extract.bib",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@ARTICLE{Kraemer:1999:GRF,
author = {Kr{\"a}mer, Walter},
title = {Gleichm{\"a}{\ss}ige ({R}undungs-){F}ehlerschranken f{\"u}r {G}leitkommaalgorithmen {\"u}ber {D}atenbereichen},
journal = {Z. angew. Math. Mech.},
fjournal = {ZAMM --- Journal of Applied Mathematics and Mechanics / Zeitschrift fur Angewandte Mathematik und Mechanik},
publisher = {Wiley-VCH Verlag},
year = {1999},
volume = {79},
number = {SUPPL. 1},
pages = {245--246},
issn = {1521-4001 (online)},
DOI = {10.1002/zamm.19990791364},
doi-URL = {http://dx.doi.org/10.1002/zamm.19990791364},
url = {http://onlinelibrary.wiley.com/doi/10.1002/zamm.19990791364/pdf},
remark = {Supplement: Minisymposia University of Bremen, Germany April 6-9, 1998;
article first published online: 18 MAR 2011},
language = {German},
keywords = "",
abstract = "",
bibdate = today,
bibsource = "http://onlinelibrary.wiley.com/documentcitationdownload?publicationDoi=10.1002/(ISSN)1521-4001&doi=10.1002/zamm.19990791364&type=journal",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@ARTICLE{Kraemer:1999:MSE,
author = {Kr{\"a}mer, Walter},
title = {Modifikationen eines {S}atzes von {E}hlich/{Z}eller zur {W}ertebereichsbestimmung bei rationalen {F}unktionen},
journal = {Z. angew. Math. Mech.},
fjournal = {ZAMM --- Journal of Applied Mathematics and Mechanics / Zeitschrift fur Angewandte Mathematik und Mechanik},
publisher = {Wiley-VCH, Weinheim},
year = {1999},
volume = {79},
number = {SUPPL. 3},
pages = {s865--s866},
ISSN = {0044-2267; 1521-4001/e},
remark = {Annual meeting, Gesellschaft f{\"u}r Angewandte Mathematik und Mechanik: GAMM 98},
language = {German},
MRCLASS = {65G30, 65D15, 41A20},
keywords = "range enclosure for rational functions; error bounds; rational approximation; interval arithmetic",
abstract = "In ihrem Artikel [Math. Z. 86, 41-44 (1964; Zbl 0131.06602)] gehen H. Ehlich
und K. Zeller der Frage nach, wie weit ein Polynom {\"u}ber einem vorgegebenen Bereich
maximal von der Nullfunktion abweichen kann. Sie geben Verfahren an, die es erlauben,
aus Betr{\"a}gen von Polynomwerten an gewissen St{\"u}tzstellen eine kontinuierlich g{\"u}ltige
Oberschranke dieser Maximalabweichung zu bestimmen. Bei Fehlerabsch{\"a}tzungen im
Zusammenhang mit der rationalen Approximation von Funktionen stellt sich die allgemeinere
Aufgabe, die maximale Abweichung des Quotienten eines Intervallpolynoms
durch ein Punktpolynom {\"u}ber einem vorgegebenen Bereich zu bestimmen. Hierzu werden
in der vorliegenden Arbeit die Ans{\"a}tze von Ehlich und Zeller [loc. cit.] geeignet verallgemeinert.",
bibdate = today,
bibsource = "https://zbmath.org/?q=an:0973.65035; https://zbmath.org/bibtex/01355089.bib",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@ARTICLE{Bantle:2000:ABR,
author = {Bantle, A. and Kr{\"a}mer, Walter},
title = {Automatische {B}estimmung von relativen worst case {F}ehlerschranken},
journal = {Z. angew. Math. Mech.},
fjournal = {Zeitschrift f{\"u}r Angewandte Mathematik und Mechanik (ZAMM)},
publisher = {Wiley-VCH, Weinheim},
year = {2000},
volume = {80},
number = {},
pages = {S819--S820},
ISSN = {0044-2267; 1521-4001/e; 0946-8463},
language = {German},
keywords = {},
abstract = {},
remark = {Annual meeting, Gesellschaft f{\"u}r Angewandte Mathematik und Mechanik; GAMM 1999},
bibdate = today,
bibsource = "https://getinfo.de/app/subject-search?",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@Article{Hofschuster:2000:MFS,
author = "Werner Hofschuster and Walter Kr{\"a}mer",
title = {Mathematical Function Software on the Web -- Are Such
Codes Useful for Verification Algorithms?},
JOURNAL = "Reliab. Comput.",
fjournal = "Reliable Computing",
year = "2000",
number = "2",
volume = "6",
pages = "207--218",
ISSN = {1385-3139 (print), 1573-1340 (online)},
CODEN = {RCOMF8},
DOI = {10.1023/A:1009973407908},
URL = {http://dx.doi.org/10.1023/A:1009973407908},
doi-URL = {http://dx.doi.org/10.1023/A:1009973407908},
MRCLASS = "65G20, 65-04",
keywords = "",
abstract = {We give an overview on existing software implementations of special mathematical functions (erf(x), erfe(x),
$\Gamma (x)$, Bessel-Functions, ...) which can be found on the web. We discuss the quality of the numerical results and their usability in an interval setting. We also point out whether it is an easy or difficult task to find reliable routines with approved (relative or absolute) error bounds. We will show which additional steps have to be performed to get worst-case error bounds for such routines.},
bibdate = today,
bibsource = "http://rd.springer.com/export-citation/article/10.1023/A%3A1009973407908.bib;
http://liinwww.ira.uka.de/bibliography//Misc/RC.html;
http://dblp.uni-trier.de/db/journals/rc/rc6.html#HofschusterK00;
http://www.ams.org/mathscinet/ MRNUMBER=1769114",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@TechReport{Hofschuster:2000:RNA,
author = {Hofschuster, Werner and Kr{\"a}mer, Walter},
title = {Rechnerisches {N}achvollziehen und {A}nalyse der praktischen {U}msetzung des {F}inanzausgleichsgesetzes},
type = "Preprint",
number = "BUW-WRSWT 2000/1",
year = "2000",
institution = "Bergische Universit{\"a}t Wuppertal",
address = "Wuppertal, Germany",
url = "http://www2.math.uni-wuppertal.de/~xsc/preprints/prep_00_1.pdf",
language = {German},
keywords = "",
abstract = "",
bibdate = today,
bibsource = "",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@ARTICLE{Kaucher:2000:LUL,
author = {Kaucher, Edgar and Kr{\"a}mer, Walter},
title = {L{\"o}sungseinschlie{\ss}ung bei unendlichen linearen {G}leichungssystemen},
journal = {Z. angew. Math. Mech.},
fjournal = {Zeitschrift f{\"u}r Angewandte Mathematik und Mechanik (ZAMM)},
publisher = {Wiley-VCH, Weinheim},
year = {2000},
volume = {80},
number = {SUPPL. 3},
pages = {S823-S824},
language = {German},
MRCLASS = {65F30; 65G30; 65G20},
keywords = {},
abstract = {},
remark = {({B}oundaries for solutions of infinite linear systems of equations), Annual meeting, Gesellschaft f{\"u}r Angewandte Mathematik und Mechanik; GAMM 99},
bibdate = today,
bibsource = "https://zbmath.org/?q=an:0960.65051; https://zbmath.org/bibtex/01497156.bib",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@TechReport{Kraemer:2000:AGE,
author = {Kr{\"a}mer, Walter and Blomquist, F.},
title = {Algorithms with Guaranteed Error Bounds for the Error Function and the Complementary Error Function},
type = "Preprint",
number = "BUW-WRSWT 2000/2",
year = "2000",
institution = "Bergische Universit{\"a}t Wuppertal",
address = "Wuppertal, Germany",
url = "http://www2.math.uni-wuppertal.de/~xsc/preprints/prep_00_2.pdf",
language = {English},
keywords = "error function, complementary error function, reliable error estimates, IEEE-double format, special functions",
abstract = "The given algorithms allow the calculation of confinements for values of the error function
erf(x) resp. the complementary error function erfc(x) for point and interval arguments
in the IEEE-double number format [29]. As well the approximation errors in
the various parts as all appearing rounding errors are seized certainly with a priori error
estimates by using interval methods. The obtained worst case error bounds for the
maximum relative error are valid for all admissible arguments simultaneously. They
are finally used for certain confinement of the ranges over points or intervals. Under
the address http://www.math.uni-wuppertal.de/wrswt/software/erf a complete XSC
implementation [25] of the algorithms discussed can be found. All approximation coefficients
are given, such that a translation into another programming language is very simple.",
bibdate = today,
bibsource = "",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@TechReport{Brauer:2001:SCX,
author = "Br{\"a}uer, M. and Hofschuster, Werner and Kr{\"a}mer, Walter",
title = {Steigungsarithmetiken in {C}-{XSC}},
type = "Preprint",
number = "BUW-WRSWT 2001/3",
year = "2001",
institution = "Bergische Universit{\"a}t Wuppertal",
address = "Wuppertal, Germany",
url = "http://www2.math.uni-wuppertal.de/~xsc/preprints/prep_01_3.pdf",
language = {German},
keywords = "Automatic slope generation, slopes, forward method, backward method, selfverifying algorithms, C-XSC",
abstract = "Slope arithmetics in C-XSC: An introduction to slope arithmetics in forward and reverse mode is
given. Implementations in C-XSC are described. Sample programs illustrate how to use the software
package.",
bibdate = today,
bibsource = "",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@TechReport{Hofschuster:2001:CCL,
author = "Hofschuster, Werner and Kr{\"a}mer, Walter and Wedner, S. and Wiethoff, A.",
title = {C-{XSC} 2.0 --- A {C}++ Class Library for Extended Scientific Computing},
type = "Preprint",
number = "BUW-WRSWT 2001/1",
year = "2001",
institution = "Bergische Universit{\"a}t Wuppertal",
address = "Wuppertal, Germany",
url = "http://www2.math.uni-wuppertal.de/~xsc/preprints/prep_01_1.pdf",
language = {English},
keywords = "",
abstract = "C-XSC is a tool for the development of numerical algorithms delivering highly accurate and automatically verified results. It provides a large number of predefined numerical data types and operators. These types are
implemented as C++ classes. Thus, C-XSC allows high-level programming of numerical applications
in C++. The C-XSC package is available for many computers with a C++ compiler conforming to
the C++ standard [12]. The sources of the new version C-XSC 2.0 as well as the problem solving
routines [3] are freely available. C-XSC 2.0 now conforms ISO/IEC C++ standard [12].",
bibdate = today,
bibsource = "",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@article{Kraemer:2001:AFE,
AUTHOR = {Kr{\"a}mer, Walter and Bantle, Armin},
TITLE = {Automatic forward error analysis for floating point algorithms},
JOURNAL = {Reliab. Comput.},
FJOURNAL = {Reliable Computing},
VOLUME = {7},
YEAR = {2001},
NUMBER = {4},
PAGES = {321--340},
ISSN = {1385-3139 (print), 1573-1340 (online)},
CODEN = {RCOMF8},
DOI = {10.1023/A:1011463324243},
URL = {http://dx.doi.org/10.1023/A:1011463324243},
doi-URL = {http://dx.doi.org/10.1023/A:1011463324243},
MRCLASS = {65G40, 65G50},
keywords = {},
abstract = {We investigate absolute and relative error bounds for floating point calculations determined by means of sequences of instructions (as, for example, given by a computer program). We get rigorous error bounds on the round-off or generated error due to the actual machine floating point operations, as well as the propagated error from one sequence to the next in a very convenient way by the computer itself. The results stated in the theorems can be used to implement software tools for the automatic computation of a priori worst case error bounds for floating point computations. These automatically computed bounds are valid simultaneously for all data vectors varying in the domain specified and their corresponding machine vectors fulfilling a maximum prescribed error bound. With great success we have used our method in the past to implement a fast interval library for elementary functions called FI\_LIB [12]. Further numerical examples often show a high quality of the computed a priori bounds.},
bibdate = today,
bibsource = "http://rd.springer.com/export-citation/article/10.1023/A%3A1011463324243.bib;
http://www.ams.org/mathscinet/, MRNUMBER=1843023 (2002d:65046)",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@TechReport{Kraemer:2001:ICM,
author = "Kr{\"a}mer, Walter and Geulig, I.",
title = {Interval Calculus in {Maple} --- The Extension {intpakX} to the Package {intpak} of the Share-Library},
type = "Preprint",
number = "BUW-WRSWT 2001/2",
year = "2001",
institution = "Bergische Universit{\"a}t Wuppertal",
address = "Wuppertal, Germany",
url = "http://www2.math.uni-wuppertal.de/~xsc/preprints/prep_01_2.pdf",
language = {English},
keywords = "Maple V, interval analysis, validated computations, disc arithmetic,
visualization of self validating numerical algorithms",
abstract = "The package intpak of the Share-Library of Maple places a first experimental interval
arithmetic package at the user's disposal. It contains among others the new data type
interval (long number intervals), the corresponding basic arithmetic operations, lattice
operations as intersection and union, some basic (long number) interval functions as well as
a command which transforms a given expression automatically into an interval expression.
The extension intpakX supplements the package intpak in essential points. E. g. the
so-called extended interval division is provided, it contains the realization of an interval
Newton procedure, a complex disc arithmetic, an extension of the exponential function to
disc intervals as well as the realization of various algorithms for confining the range of a complex
polynomial (with centred multiplication, with area optimal multiplication). Moreover,
by providing suitable procedures the extension aims at graphically vizualizing the verification
algorithms (e. g. search for zeros, linear, quadratic range confinements, disc interval calculus).
The monitor output of the graphic routines is (in contrast to the figures in this report)
coloured.
The source code (about 2000 lines Maple code) of the extension intpakX is freely available.",
bibdate = today,
bibsource = "",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@book{Kraemer:2001:SCV,
title = {Scientific Computing, Validated Numerics, Interval Methods},
editor = {Kr{\"a}mer, Walter and {Wolff von Gudenberg}, J{\"u}rgen},
year = {2001},
publisher = {Kluwer Academic/Plenum},
address = {Boston/Dordrecht/London},
isbn = {978-1-4757-6484-0},
pages = {398},
url = {http://www.springer.com/computer/swe/book/978-0-306-46706-6},
bibdate = today,
bibsource = "",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@TechReport{Lerch:2001:ILF,
author = {Lerch, Michael and Tischler, German and {Wolff von Gudenberg},
J{\"u}rgen and Hofschuster, Werner and Kr{\"a}mer, Walter},
title = {The Interval Library filib++ 2.0 -- Design, Features and Sample Programs},
type = "Preprint",
number = "BUW-WRSWT 2001/4",
year = "2001",
institution = "Bergische Universit{\"a}t Wuppertal",
address = "Wuppertal, Germany",
url = "http://www2.math.uni-wuppertal.de/~xsc/preprints/prep_01_4.pdf",
language = {English},
keywords = "",
abstract = "",
bibdate = today,
bibsource = "",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@TechReport{Hofschuster:2002:CCL,
author = "Hofschuster, Werner and Kr{\"a}mer, Walter",
title = {C-{XSC} 2.0 --- A {C}++ Class Library for Extended Scientific Computing},
type = "Preprint",
number = "BUW-WRSWT 2002/4",
year = "2002",
institution = "Bergische Universit{\"a}t Wuppertal",
address = "Wuppertal, Germany",
url = "http://www2.math.uni-wuppertal.de/~xsc/preprints/prep_02_4.pdf",
language = {English},
keywords = "",
abstract = "",
bibdate = today,
bibsource = "",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@inproceedings{Kraemer:2002:AST,
title = {Advanced software tools for validated computing},
author = {Kr{\"a}mer, Walter},
booktitle = {Proceedings of the 31st Spring Conference of the Union of Bulgarian Mathematicians},
pages = {344--355},
year = {2002},
url = {http://www2.math.uni-wuppertal.de/wrswt/preprints/prep_02_1.pdf},
preprint-url = {http://www2.math.uni-wuppertal.de/~xsc/preprints/prep_02_1.pdf},
remark = {See also Preprint BUGHW-WRSWT 2002/1, Bergische Universit{\"a}t Wuppertal, Wuppertal, 2002.},
keywords = "",
abstract = "",
bibdate = today,
bibsource = "http://www2.math.uni-wuppertal.de/~xsc/literatur.html",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@TechReport{Miehe:2002:VRS,
author = "Miehe, D. and Kr{\"a}mer, Walter and Hofschuster, Werner",
title = {Visualization of Resulting Sets Coming from Multiplication and Division of Complex Intervals},
type = "Preprint",
number = "BUW-WRSWT 2002/3",
year = "2002",
institution = "Bergische Universit{\"a}t Wuppertal",
address = "Wuppertal, Germany",
url = "http://www2.math.uni-wuppertal.de/~xsc/preprints/prep_02_3.pdf",
language = {English},
keywords = "",
abstract = "",
bibdate = today,
bibsource = "",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@TechReport{Holbig:2003:SSD,
author = "Carlos Amaral H{\"o}lbig and Walter Kr{\"a}mer",
title = {Selfverifying Solvers for Dense Systems of Linear Equations Realized in {C}-{XSC}},
type = "Preprint",
number = "BUW-WRSWT 2003/1",
year = "2003",
institution = "Bergische Universit{\"a}t Wuppertal",
address = "Wuppertal, Germany",
url = "http://www2.math.uni-wuppertal.de/~xsc/preprints/prep_03_1.pdf",
language = {English},
keywords = "",
abstract = "",
bibdate = today,
bibsource = "",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@Article{Kraemer:2003:EIP,
author = "Walter Kr{\"a}mer and {Wolff von Gudenberg}, J{\"u}rgen",
title = {Extended Interval Power Function},
JOURNAL = {Reliab. Comput.},
fjournal = "Reliable Computing",
year = "2003",
number = "5",
volume = "9",
pages = "339--347",
issn = "1385-3139 (print), 1573-1340 (online)",
CODEN = "RCOMF8",
DOI = "10.1023/A:1025175029490",
URL = "http://dx.doi.org/10.1023/A:1025175029490",
doi-URL = "http://dx.doi.org/10.1023/A:1025175029490",
remark = "Proceedings of the Validated Computing 2002 Conference (Toronto, ON)",
MRCLASS = "65G30",
keywords = "",
abstract = "The containment set as an extension of the range of a function has been introduced in a series of white papers; see e.g. Walster, G. W.: Closed Interval Systems, Sun Microsystems, 2002, and Walster, G. W. et al.: Extended Real Intervals and the Topological Closure of Extended Real Relations, Sun Microsystems, 2002. The containment evaluation provides an exception free evaluation of functions over an arbitrary range. In this paper we discuss alternative existing implementations (C++ Interval Arithmetic Programming Reference, Sun Microsystems, 2000, and Hofschuster, W. et al.:
The Interval Library fi\_lib++ 2.0, Design, Features and Sample Programs, Universit{\"a}t Wuppertal, 2001) of the power function, introduce a new version, develop containment sets and discuss algorithms for the implementation.",
bibdate = today,
bibsource = "http://rd.springer.com/export-citation/article/10.1023/A%3A1025175029490.bib;
http://liinwww.ira.uka.de/bibliography//Misc/RC.html;
http://dblp.uni-trier.de/db/journals/rc/rc9.html#KramerG03;
http://www.ams.org/mathscinet/, MRNUMBER=2016445",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@TechReport{Popova:2003:PFI,
author = "Evgenija D. Popova and Walter Kr{\"a}mer",
title = {Parametric Fixed-Point Iteration Implemented in {C}-{XSC}},
type = "Preprint",
number = "BUW-WRSWT 2003/3",
year = "2003",
institution = "Bergische Universit{\"a}t Wuppertal",
address = "Wuppertal, Germany",
url = "http://www2.math.uni-wuppertal.de/~xsc/preprints/prep_03_3.pdf",
language = {English},
keywords = "",
abstract = "",
bibdate = today,
bibsource = "",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@TechReport{Blomquist:2004:RHC,
author = "Frithjof Blomquist and Werner Hofschuster and Walter Kr{\"a}mer",
title = {Realisierung der hyperbolischen {C}otangens-{F}unktion in einer {S}taggered {C}orrection {I}ntervallarithmetik in {C}-{XSC}},
type = "Preprint",
number = "BUW-WRSWT 2004/3",
year = "2004",
institution = "Bergische Universit{\"a}t Wuppertal",
address = "Wuppertal, Germany",
url = "http://www2.math.uni-wuppertal.de/~xsc/preprints/prep_04_3.pdf",
language = {German},
keywords = "",
abstract = "",
bibdate = today,
bibsource = "",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@TechReport{Blomquist:2004:SAA,
author = "Frithjof Blomquist and Werner Hofschuster and Walter Kr{\"a}mer",
title = {Sichere a priori {A}bsch{\"a}tzungen und {R}ealisierung der {F}unktion $\sqrt{x^2-1}$ in {C}-{XSC}},
type = "Preprint",
number = "BUW-WRSWT 2004/5",
year = "2004",
institution = "Bergische Universit{\"a}t Wuppertal",
address = "Wuppertal, Germany",
url = "http://www2.math.uni-wuppertal.de/~xsc/preprints/prep_04_5.pdf",
language = {German},
keywords = "",
abstract = "",
bibdate = today,
bibsource = "",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@TechReport{Blomquist:2004:SAF,
author = "Frithjof Blomquist and Werner Hofschuster and Walter Kr{\"a}mer",
title = {Sichere a priori {F}ehlerabsch{\"a}tzung und {I}mplementierung der {F}unktion zweier {V}ariabler
$\log (\sqrt{x^2+y^2})$},
type = "Preprint",
number = "BUW-WRSWT 2004/1",
year = "2004",
institution = "Bergische Universit{\"a}t Wuppertal",
address = "Wuppertal, Germany",
url = "http://www2.math.uni-wuppertal.de/~xsc/preprints/prep_04_1.pdf",
language = {German},
keywords = "",
abstract = "",
bibdate = today,
bibsource = "",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@incollection{Hofschuster:2004:CLE,
author = {Hofschuster, W. and Kr{\"a}mer, Walter},
title = {{C}-{XSC} 2.0 --- A {C}++ Library for Extended Scientific Computing},
booktitle = {Numerical Software with Result Verification},
editor = {Alt, Rene and Frommer, Andreas and Kearfott, R.Baker and Luther, Wolfram},
series = {Lecture Notes in Computer Science},
publisher = {Springer Berlin Heidelberg},
volume = {2991},
year = {2004},
pages = {15--35},
isbn = {978-3-540-21260-7 (print), 978-3-540-24738-8 (electronic)},
ISSN = "0302-9743",
CODEN = "LNCSD9",
doi = {10.1007/978-3-540-24738-8_2},
URL = {http://dx.doi.org/10.1007/978-3-540-24738-8_2},
doi-url = {http://dx.doi.org/10.1007/978-3-540-24738-8_2},
preprint-url = {http://www2.math.uni-wuppertal.de/~xsc/preprints/prep_04_1.pdf},
remark = {International Dagstuhl Seminar, Dagstuhl Castle, Germany, January 19-24, 2003, Revised Papers.
See also Preprint BUW-WRSWT 2003/5, Bergische Universit{\"a}t Wuppertal, Wuppertal, 2003.},
abstract = {In this note the main features and newer developments of the C++ class library for extended scientific computing C-XSC 2.0 will be discussed. The original version of the C-XSC library is about ten years old. But in the last decade the underlying programming language C++ has been developed significantly. Since November 1998 the C++ standard is available and more and more compilers support (most of) the features of this standard. The new version C-XSC 2.0 conforms to this standard. Application programs written for older C-XSC versions have to be modified to run with C-XSC 2.0. Several examples will help the user to see which changes have to be done. Note, that all sample codes given in [6] have to be modified to work properly with C-XSC 2.0. All sample codes listed in this note will be made available on the web page \url{http://www.math.uni-wuppertal.de/~xsc/cxsc/examples}.},
bibdate = today,
bibsource = "http://rd.springer.com/export-citation/chapter/10.1007/978-3-540-24738-8_2.bib;
http://liinwww.ira.uka.de/bibliography//Misc/LNCS/index.html;
http://dblp.uni-trier.de/db/conf/dagstuhl/ns2003.html#HofschusterK03",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@InCollection{Holbig:2004:AES,
author = "Carlos Amaral H{\"o}lbig and Walter Kr{\"a}mer and Tiaraju Asmuz Diverio",
title = {An Accurate an Efficient Selfverifying Solver for
Systems with Banded Coefficient Matrix},
booktitle = "Parallel Computing: Software Technology, Algorithms,
Architectures and Applications, PARCO'03",
editor = "Gerhard R. Joubert and Wolfgang E. Nagel and Frans J. Peters and W. V. Walter",
series = "Advances in Parallel Computing",
volume = "13",
pages = "283--290",
publisher = "Elsevier (Amsterdam, The Netherlands)",
year = "2004",
ISBN = "0-444-51689-1",
remark = {Proceedings of ParCo2003, Dresden, Germany, 2003.},
bibdate = today,
bibsource = "Holbig04 in http://liinwww.ira.uka.de/bibliography//Parallel/Eugene/index.html;
http://dblp.uni-trier.de/db/conf/parco/parco2003.html#HolbigKD03",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@article{Kraemer:2004:ZBV,
AUTHOR = {Kr{\"a}mer, Walter and Popova, Evgenija D.},
TITLE = {Zur {B}erechnung von verl{\"a}sslichen {A}u{\ss}en- und {I}nneneinschlie{\ss}ungen bei parameterabh{\"a}ngigen linearen {G}leichungssystemen},
JOURNAL = {Proc. Appl. Math. Mech. (PAMM)},
FJOURNAL = {Proceedings in Applied Mathematics and Mechanics},
VOLUME = {4},
YEAR = {2004},
NUMBER = {1},
PAGES = {670--671},
PUBLISHER = {WILEY-VCH Verlag},
ISSN = {1617-7061},
DOI = {10.1002/pamm.200410316},
doi-URL = {http://dx.doi.org/10.1002/pamm.200410316},
URL = {http://onlinelibrary.wiley.com/doi/10.1002/pamm.200410316/pdf},
KEYWORDS = {},
ABSTRACT = {Es wird ein selbstverifizierendes Verfahren zur Berechnung von Innen- und Au{\ss}eneinschlie{\ss}ungen der L{\"o}sungsmenge eines parameterabh{\"a}ngigen linearen Gleichungssystems bei affin-linearer Parameterabh{\"a}ngigkeit besprochen. Entsprechende open-source-Software wird online zur Verf{\"u}gung gestellt.},
bibdate = today,
bibsource = "http://onlinelibrary.wiley.com/documentcitationdownload?publicationDoi=10.1002/(ISSN)1617-7061&doi=10.1002/pamm.200410316&type=journal",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@TechReport{Blomquist:2006:CAE,
author = {Blomquist, F. and Kr{\"a}mer, Walter},
title = {Computing a priori Error Bounds for Floating-point Evaluations of Arithmetic Expressions},
type = "Preprint",
number = "BUW-WRSWT 2006/1",
year = "2006",
institution = "Bergische Universit{\"a}t Wuppertal",
address = "Wuppertal, Germany",
url = "http://www2.math.uni-wuppertal.de/~xsc/preprints/prep_06_1.pdf",
keywords = "",
abstract = "",
bibdate = today,
bibsource = "",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@TechReport{Blomquist:2005:CIF,
author = "Frithjof Blomquist and Werner Hofschuster and Walter Kr{\"a}mer and Neher, Markus",
title = {Complex Interval Functions in {C-XSC}},
type = "Preprint",
number = "BUW-WRSWT 2005/2",
year = "2005",
institution = "Bergische Universit{\"a}t Wuppertal",
address = "Wuppertal, Germany",
url = "http://www2.math.uni-wuppertal.de/~xsc/preprints/prep_05_2.pdf",
language = {English},
keywords = "",
abstract = "",
bibdate = today,
bibsource = "",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@TechReport{Blomquist:2005:RCC,
author = "Frithjof Blomquist and Werner Hofschuster and Walter Kr{\"a}mer",
title = {Reliable Computation of the Complex Interval Function $\arcsin(z)$ in {C-XSC}},
type = "Preprint",
number = "BUW-WRSWT 2005/1",
year = "2005",
institution = "Bergische Universit{\"a}t Wuppertal",
address = "Wuppertal, Germany",
url = "http://www2.math.uni-wuppertal.de/~xsc/preprints/prep_05_1.pdf",
language = {English},
keywords = "",
abstract = "",
bibdate = today,
bibsource = "",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@TechReport{Blomquist:2005:RCT,
author = "Frithjof Blomquist and Werner Hofschuster and Walter Kr{\"a}mer",
title = {Real and Complex {T}aylor Arithmetic in {C-XSC}},
type = "Preprint",
number = "BUW-WRSWT 2005/4",
year = "2005",
institution = "Bergische Universit{\"a}t Wuppertal",
address = "Wuppertal, Germany",
url = "http://www2.math.uni-wuppertal.de/~xsc/preprints/prep_05_4.pdf",
language = {English},
keywords = "",
abstract = "",
bibdate = today,
bibsource = "",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@article {Kraemer:2006:GID,
AUTHOR = {Kr{\"a}mer, Walter},
TITLE = {Generalized Intervals and the Dependency Problem},
JOURNAL = {Proc. Appl. Math. Mech. (PAMM)},
FJOURNAL = {Proceedings in Applied Mathematics and Mechanics},
VOLUME = {6},
YEAR = {2006},
NUMBER = {1},
PAGES = {683--684},
PUBLISHER = {WILEY-VCH Verlag},
ISSN = {1617-7061},
DOI = {10.1002/pamm.200610322},
URL = {http://onlinelibrary.wiley.com/doi/10.1002/pamm.200610322/pdf},
DOI-URL = {http://dx.doi.org/10.1002/pamm.200610322},
preprint-url = {http://www2.math.uni-wuppertal.de/~xsc/preprints/prep_06_4.pdf},
remark = "Special Issue: GAMM Annual Meeting 2006, Berlin",
KEYWORDS = {},
ABSTRACT = {Computing enclosures of the range of functions using interval arithmetic often leads to overestimations due to variables (parameters) appearing more than once within the expression to be evaluated (dependency problem). An experimental implementation of a generalized interval arithmetic which has been proposed by Hansen [2] in 1975 is now available [1, 3]. In many cases generalized intervals are well suited to reduce the dependency problem and/or the so called wrapping effect.},
bibdate = today,
bibsource = {http://onlinelibrary.wiley.com/documentcitationdownload?publicationDoi=10.1002/(ISSN)1617-7061&doi=10.1002/pamm.200610322&type=journal},
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@TechReport{Kraemer:2006:PIM,
author = {Kr{\"a}mer, Walter},
title = {Pitfalls in {M}aple},
type = "Preprint",
number = "BUW-WRSWT 2006/7",
year = "2006",
institution = "Bergische Universit{\"a}t Wuppertal",
address = "Wuppertal, Germany",
keywords = "",
abstract = "",
bibdate = today,
bibsource = "http://www2.math.uni-wuppertal.de/~xsc/literatur.html",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@article{Lerch:2006:FFI,
AUTHOR = {Lerch, Michael and Tischler, German and {Wolff von Gudenberg},
J{\"u}rgen and Hofschuster, Werner and Kr{\"a}mer, Walter},
TITLE = {F{ILIB}++, a fast interval library supporting containment computations},
JOURNAL = {ACM Trans. Math. Software},
FJOURNAL = {Association for Computing Machinery. Transactions on Mathematical Software},
VOLUME = {32},
month = {June},
YEAR = {2006},
NUMBER = {2},
PAGES = {299--324},
CODEN = "ACMSCU",
ISSN = "0098-3500 (print), 1557-7295 (electronic)",
issn-l = "0098-3500",
DOI = "10.1145/1141885.1141893",
URL = "http://dx.doi.org/10.1145/1141885.1141893",
doi-URL = "http://dx.doi.org/10.1145/1141885.1141893",
preprint-url = "http://www2.math.uni-wuppertal.de/~xsc/preprints/prep_03_4.pdf",
journal-url = "http://dl.acm.org/pub.cfm?id=J782",
MRCLASS = {65G30},
keywords = {C++ class library; containment computations; containment sets; exception free computations; Filib++; guaranteed numerical results; interval arithmetic; interval computations; templates; traits class; validated numerics; validation},
abstract = {filib++ is an extension of the interval library filib originally developed at the University of Karlsruhe. The most important aim of filib is the fast computation of guaranteed bounds for interval versions of a comprehensive set of elementary functions. filib++ extends this library in two aspects. First, it adds a second mode, the extended mode, that extends the exception-free computation mode (using special values to represent infinities and NaNs known from the IEEE floating-point standard 754) to intervals. In this mode, the so-called containment sets are computed to enclose the topological closure of a range of a function over an interval. Second, our new design uses templates and traits classes to obtain an efficient, easily extendable, and portable C++ library.},
bibdate = today,
bibsource = "http://www.acm.org/pubs/contents/journals/toms/;
http://www.math.utah.edu/pub/tex/bib/toms.bib/;
http://dblp.uni-trier.de/db/journals/toms/toms32.html#LerchTGHK06;
http://www.ams.org/mathscinet/, MRNUMBER=2272364 (2007k:65075)",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@TechReport{Blomquist:2007:VUU,
author = "Frithjof Blomquist and Werner Hofschuster and Walter Kr{\"a}mer",
title = {Vermeidung von {\"U}ber- und {U}nterlauf und {V}erbesserung der {G}enauigkeit
bei reeller und komplexer staggered {I}ntervall-{A}rithmetik},
type = "Preprint",
number = "BUW-WRSWT 2007/8",
year = "2007",
institution = "Bergische Universit{\"a}t Wuppertal",
address = "Wuppertal, Germany",
url = "http://www2.math.uni-wuppertal.de/~xsc/preprints/prep_07_8_na.pdf",
language = {German},
keywords = "",
abstract = "",
remark = "talk at a Dagstuhl seminar",
bibdate = today,
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@InProceedings{Grimmer:2007:MEV,
author = "Markus Grimmer and Walter Kr{\"a}mer",
title = {An {MPI} Extension for Verified Numerical Computations in Parallel Environments},
booktitle = "Proceedings of the 2007 International Conference on
Scientific Computing, {CSC} 2007, June 25-28, 2007, Las Vegas, Nevada, {USA}",
editor = "Hamid R. Arabnia and Jack Y. Yang and Mary Qu Yang",
publisher = "CSREA Press",
year = "2007",
pages = "111--117",
ISBN = "1-60132-037-X",
bibdate = today,
bibsource = "http://dblp.uni-trier.de/db/conf/csc/csc2007.html#GrimmerK07",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@inproceedings{Kraemer:2007:ACH,
AUTHOR = {Kr{\"a}mer, Walter},
TITLE = {Accurate computation of chaotic dynamical systems},
BOOKTITLE = {Proceedings of the 8th WSEAS Int. Conf. on Mathematics and Computers in Biology and Chemistry (MCBC '07)},
EDITOR = {Akshai Aggarwal},
PUBLISHER = {WSEAS},
issn = {1790-5095},
isbn = {978-960-8457-75-1 },
year = {2007},
pages = { 74--79},
URL = {http://www.wseas.us/e-library/conferences/2007vancouver/papers/558-240.pdf},
keywords = {dynamical system, logistic equation, chaotic behaviour, verified numerical results, intpakX},
abstract = {The computation of orbits of dynamical systems is known to be highly unstable if the system exhibits
chaotic behavior. In this case, even for the very simplest systems, ordinary floating-point computations will eventually
deliver results which are completely wrong quantitatively, when compared with the true trajectory on which
the computation began. Similarly, ordinary interval arithmetic (i. e. intervals of floating-point numbers) yield poor
enclosures after few iterations. In most cases the computation breaks down because of overflow. Using intpakX's
multiple precision intervals, we can compute enclosures of orbits for a considerably longer time with high accuracy.
Statements concerning the sensitivity with respect to small changes in the seed value of the numerical computations
are possible with mathematical rigor. We also show that computing an orbit using a rational arithmetic as
e.g. provided by the computer algebra system Maple is not possible due to computing time and computer memory limitations.},
bibdate = today,
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@article{Kraemer:2007:BEU,
author = {Kr{\"a}mer, Walter},
title = {Bugs, errors, and unexpected results in computer algebra packages},
journal = {Proc. Appl. Math. Mech. (PAMM)},
fjournal = {Proceedings in Applied Mathematics and Mechanics},
year = {2007},
volume = {7},
number = {1},
pages = {2140009--2140010},
publisher = {WILEY-VCH Verlag},
issn = {1617-7061},
DOI = {10.1002/pamm.200700756},
url = {http://dx.doi.org/10.1002/pamm.200700756},
doi-url = {http://dx.doi.org/10.1002/pamm.200700756},
remark = {Special Issue: Sixth International Congress on Industrial Applied Mathematics (ICIAM07) and GAMM Annual Meeting, Z{\"u}rich 2007.},
abstract = {The following is a citation from the description and aims of the Dagstuhl Seminar on Algebraic and Numerical Algorithms and Computer-assisted Proofs:
In contrast, most algorithms in computer algebra are "never failing", that is they are proved to provide a solution for any input, and the maximum computing time for this is estimated a priori.
The focus of this contribution is the reliability of the computer algebra system Maple. We will show that "never failing" algorithms may be the theory but concerning actual implementations such a statement seems not to be drawn from life.},
bibdate = today,
bibsource = "http://onlinelibrary.wiley.com/documentcitationdownload?publicationDoi=10.1002/(ISSN)1617-7061&doi=10.1002/pamm.200700756&type=journal",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@article{Kraemer:2007:CVS,
AUTHOR = {Kr{\"a}mer, Walter},
TITLE = {Computing and visualizing solution sets of interval linear systems},
JOURNAL = {Serdica J. Comput.},
FJOURNAL = {Serdica Journal of Computing},
VOLUME = {1},
YEAR = {2007},
NUMBER = {4},
PAGES = {455--468},
ISSN = {1312-6555 (print), 1314-7897 (online)},
URL = {http://serdica-comp.math.bas.bg/index.php/serdicajcomputing/article/download/33/30},
preprint-url = {http://www2.math.uni-wuppertal.de/~xsc/preprints/prep_06_8.pdf},
MRCLASS = {65F05, 65G40},
KEYWORDS = {solution sets; interval linear systems; reliable computations; visualization using computer algebra tools; intpakX},
ABSTRACT = {The computation of the exact solution set of an interval linear system is a nontrivial task [2, 13]. Even in two and three dimensions a lot of work has to be done. We demonstrate two different realizations. The first approach (see [16]) is based on Java, Java3D, and the BigRational package [21]. An applet allows modifications of the matrix coefficients and/or the coefficients of the right hand side with concurrent real time visualization of the corresponding solution sets. The second approach (see [5]) uses Maple and intpakX [22, 8, 12] to implement routines for the computation and visualization of two and three dimensional solution sets. The regularity of the interval matrix $A$ is verified by showing that $\varrho (|I- mid^{-1}(A)*Aj|) < 1$, [14]. Here, I means the identity matrix, mid$(A)$ denotes the midpoint matrix and $\varrho$ denotes the spectral radius of a real matrix.},
remark = {See also Preprint BUW-WRSWT 2006/8, Bergische Universit{\"a}t, Wuppertal, 2006.},
bibdate = today,
bibsource = "http://serdica-comp.math.bas.bg/index.php/serdicajcomputing/rt/captureCite/33/30/;
http://www.ams.org/mathscinet/, MRNUMBER=2412184 (2009e:65050)",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@inproceedings{Kraemer:2007:IIA,
AUTHOR = {Kr{\"a}mer, Walter},
TITLE = {{i}ntpak{X} --- an interval arithmetic package for {M}aple},
BOOKTITLE = {Scientific Computing, Computer Arithmetic and Validated Numerics, 2006. SCAN 2006. 12th GAMM - IMACS International Symposium on},
PUBLISHER = {IEEE Computer Society Press},
YEAR = {2007},
DATE = {26-29 Sept. 2006},
ISBN = {978-0-7695-2821-2},
NOTE = {article number 4402417},
DOI = {10.1109/SCAN.2006.29},
URL = {http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4402417},
doi-url = {http://dx.doi.org/10.1109/SCAN.2006.29},
ABSTRACT = {intpakX is a Maple package for (multiple precision) interval arithmetic. It contains data types, basic arithmetic and standard functions for real interval arithmetic and complex disc arithmetic.
Moreover, it implements a handful of algorithms for validated numerical computing and graphical output functions for the visualization of results. The package intpakX thus gives the user the opportunity to do validated computing with a Computer Algebra System.
The package may be used to solve (set valued) numerical problems in a computer algebra environment with mathematical rigor. But it is also a valuable didactical tool which allows the illustration of various interval methods/algorithms. The latest version of intpakX offers the possibility to compute and to visualize the exact solution set of two and three dimensional linear interval systems [3]. This new feature will be discussed in some detail.},
bibdate = today,
bibsource = "http://ieeexplore.ieee.org/",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@article {Kraemer:2007:IMP,
AUTHOR = {Kr{\"a}mer, Walter},
TITLE = {Introduction to the {Maple} Power Tool {intpakX}},
JOURNAL = {Serdica J. Comput.},
FJOURNAL = {Serdica Journal of Computing, Bulgarian Academy of Sciences},
VOLUME = {1},
YEAR = {2007},
NUMBER = {4},
PAGES = {469--504},
ISSN = {1312-6555 (print), 1314-7897 (online)},
CODEN = "????",
URL = {http://serdica-comp.math.bas.bg/index.php/serdicajcomputing/article/download/34/31},
preprint-url = {http://www2.math.uni-wuppertal.de/~xsc/preprints/prep_06_9.pdf},
KEYWORDS = {computer algebra; validated computations; visualization of interval methods; didactical tool; Maple Power Tool; intpakX},
ABSTRACT = {The Maple Power Tool intpakX [24] de nes Maple types for real intervals and complex disc intervals. On the level of basic operations, intpakX includes the four basic arithmetic operators, including extended interval division as an extra function. Furthermore, there are power, square, square root, logarithm and exponential functions, a set of standard functions, union, and intersection. Reimplementations of the Maple construction, conversion, and unapplication functions are available. Additionally, there is a range of operators for complex disc arithmetic. As applications, verified computation of zeroes (Interval Newton Method) with the possibility to find all zeroes of a function on a specified interval, and range enclosure for real-valued functions of one or two variables are implemented, the latter using either interval evaluation or evaluation via the mean value form and adaptive subdivision of intervals. The user can choose between a non-graphical and a graphical version of the above algorithms displaying the resulting intervals of each iteration step. The source code (about 2000 lines of Maple code) of the extension intpakX is freely available [23].},
remark = {See also Preprint BUW-WRSWT 2006/9, Bergische Universit{\"a}t Wuppertal, Wuppertal, 2006.},
bibdate = today,
bibsource = "http://serdica-comp.math.bas.bg/index.php/serdicajcomputing/rt/captureCite/34/31/;
http://www.math.utah.edu/pub/tex/bib/maple-extract.bib",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@article {Popova:2007:IOB,
AUTHOR = {Popova, Evgenija D. and Kr{\"a}mer, Walter},
TITLE = {Inner and outer bounds for the solution set of parametric linear systems},
JOURNAL = {J. Comput. Appl. Math.},
FJOURNAL = {Journal of Computational and Applied Mathematics},
VOLUME = {199},
YEAR = {2007},
NUMBER = {2},
PAGES = {310--316},
ISSN = {0377-0427},
CODEN = {JCAMDI},
DOI = {10.1016/j.cam.2005.08.048},
URL = {http://dx.doi.org/10.1016/j.cam.2005.08.048},
doi-URL = {http://dx.doi.org/10.1016/j.cam.2005.08.048},
preprint-url = {http://www2.math.uni-wuppertal.de/~xsc/preprints/prep_06_6.pdf},
remark = "Special Issue on Scientific Computing, Computer Arithmetic, and Validated Numerics (SCAN 2004),
Available also as Preprint BUW-WRSWT 2006/6, Bergische Universit{\"a}t Wuppertal, Wuppertal, 2006.",
MRCLASS = {15A06, 65G40},
KEYWORDS = {C-XSC; parametric linear systems; inner estimation; validated software},
ABSTRACT = {Consider linear systems involving linear dependencies between interval parameters. Presented is a free C-XSC software implementing a generalized parametric fixed-point iteration method for verified enclosure of the parametric solution set. Some specific features of the corresponding algorithm concerning sharp enclosure of the contracting matrix and inner approximation of the solution enclosure are discussed.},
bibdate = today,
bibsource = "http://www.ams.org/mathscinet/, MRNUMBER=2269513",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@InProceedings{Blomquist:2008:MSC,
title = {A Modified Staggered Correction Arithmetic with Enhanced Accuracy and Very Wide Exponent Range},
author = "Frithjof Blomquist and Werner Hofschuster and Walter Kr{\"a}mer",
booktitle = "Numerical Validation in Current Hardware Architectures, 6.1. - 11.1.2008",
editor = "Annie A. M. Cuyt and Walter Kr{\"a}mer and Wolfram Luther and Peter W. Markstein",
series = "Dagstuhl Seminar Proceedings",
publisher = "Internationales Begegnungs- und Forschungszentrum f{\"u}r Informatik (IBFI), Schloss Dagstuhl, Germany",
year = "2008",
volume = "08021",
ISSN = "1862-4405",
URL = "http://drops.dagstuhl.de/opus/volltexte/2008/1445/pdf/08021.KraemerWalter.Paper.1445.pdf",
keywords = "Staggered correction, multiple precision, C-XSC, interval computation, wide exponent range,
reliable numerical computations, complex interval functions",
note = {see also \cite{Blomquist:2009:MSC}},
bibdate = today,
bibsource = "http://drops.dagstuhl.de/opus/volltexte/2008/1445;
http://liinwww.ira.uka.de/bibliography//Misc/Dagstuhl_Publishing.html;
http://dblp.uni-trier.de/db/conf/dagstuhl/P8021.html#BlomquistHK08",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@TechReport{Blomquist:2008:RCS,
author = "Frithjof Blomquist and Werner Hofschuster and Walter Kr{\"a}mer",
title = {Real and Complex Staggered (Interval) Arithmetics with Wide Exponent Range (in {G}erman)},
type = "Preprint",
number = "BUW-WRSWT 2008/1",
year = "2008",
institution = "Bergische Universit{\"a}t Wuppertal",
address = "Wuppertal, Germany",
url = "http://www2.math.uni-wuppertal.de/~xsc/preprints/prep_08_1.pdf",
language = "German",
keywords = "",
abstract = "",
bibdate = today,
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@InProceedings{Cuyt:2008:SNV,
title = {08021 {S}ummary --- {N}umerical {V}alidation in {C}urrent {H}ardware {A}rchitectures},
author = "Annie A. M. Cuyt and Walter Kr{\"a}mer and Wolfram Luther and Peter W. Markstein",
booktitle = "Numerical Validation in Current Hardware Architectures, 6.1. - 11.1.2008",
editor = "Annie A. M. Cuyt and Walter Kr{\"a}mer and Wolfram Luther and Peter W. Markstein",
series = "Dagstuhl Seminar Proceedings",
publisher = "Internationales Begegnungs- und Forschungszentrum
f{\"u}r Informatik (IBFI), Schloss Dagstuhl, Germany",
year = "2008",
volume = "08021",
ISSN = "1862-4405",
URL = "http://drops.dagstuhl.de/opus/volltexte/2008/1433",
other-URL = "http://drops.dagstuhl.de/portals/08021/",
note = "see also \cite{Cuyt:2009:NVC}",
keywords = "Computer arithmetic, arbitrary precision,
floating-point arithmetic standardization, language support, reliable libraries",
bibdate = today,
bibsource = "http://dblp.uni-trier.de/db/conf/dagstuhl/P8021.html#CuytKLM08,
http://liinwww.ira.uka.de/bibliography//Misc/Dagstuhl_Publishing.html",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@InProceedings{Grimmer:2008:OSP,
author = "Markus Grimmer and Walter Kr{\"a}mer",
title = {An Open Source Parallel Interval Solver for Systems of
Linear {F}redholm Integral Equations of the Second Kind},
booktitle = "Proceedings of the 2008 International Conference on
Scientific Computing, {CSC} 2008, July 14-17, 2008, Las Vegas, Nevada, {USA}",
editor = "Hamid R. Arabnia",
publisher = "CSREA Press",
ISBN = "1-60132-059-0",
year = "2008",
pages = "204--210",
keywords = {C-XSC; Integral Equation; MPI; Open source; Selverifying numerical solver},
abstract = {We present a parallel verified C-XSC solver for linear Fredholm integral equations of the second kind based on related interval methods due to Klein [9, 10]. The serial and parallel methods are described, and test results from the new C-XSC implementation on the supercomputer AL-iCEnext in Wuppertal, Germany, are given. At the same time, new and enhanced software components in C-XSC as an MPI interface for C-XSC data types and a parallel verified linear system solver, as well as the solution visualization via the Maple package intpakX are presented.},
bibdate = today,
bibsource = "http://dblp.uni-trier.de/db/conf/csc/csc2008.html#GrimmerK08",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@InProceedings{Hofschuster:2008:CCR,
author = "Werner Hofschuster and Walter Kr{\"a}mer and Markus Neher",
title = {{C}-{XSC} and Closely Related Software Packages},
booktitle = "Numerical Validation in Current Hardware Architectures, 6.1. - 11.1.2008",
editor = "Annie Cuyt and Walter Kr{\"a}mer and Wolfram Luther and Peter Markstein",
series = "Dagstuhl Seminar Proceedings",
publisher = "Internationales Begegnungs- und Forschungszentrum f{\"u}r Informatik (IBFI), Schloss Dagstuhl, Germany",
year = "2008",
number = "08021",
ISSN = "1862-4405",
URL = "http://drops.dagstuhl.de/opus/volltexte/2008/1442/pdf/08021.HofschusterWerner.Paper.1442.pdf",
keywords = "mathematical software, reliable computing, C-XSC, CoStLy, ACETAF",
note = {see also \cite{Hofschuster:2009:CCR}},
bibdate = today,
bibsource = "http://drops.dagstuhl.de/opus/volltexte/2008/1442;
http://liinwww.ira.uka.de/bibliography//Misc/Dagstuhl_Publishing.html;
http://dblp.uni-trier.de/db/conf/dagstuhl/P8021.html#HofschusterKN08",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@InProceedings{Kolberg:2008:NSP,
author = "Mariana Kolberg and Walter Kr{\"a}mer and Michael Zimmer",
title = {A Note on Solving Problem 7 of the {SIAM} 100-Digit Challenge Using {C}-{XSC}},
booktitle = "Numerical Validation in Current Hardware Architectures, 6.1. - 11.1.2008",
editor = "Annie Cuyt and Walter Kr{\"a}mer and Wolfram Luther and Peter Markstein",
year = "2008",
number = "08021",
series = "Dagstuhl Seminar Proceedings",
ISSN = "1862-4405",
publisher = "Internationales Begegnungs- und Forschungszentrum f{\"u}r Informatik (IBFI), Schloss Dagstuhl, Germany",
URL = "http://drops.dagstuhl.de/opus/volltexte/2008/1447/pdf/08021.LuederitzKolbergMariana.Paper.1447.pdf",
preprint-url = "http://www2.math.uni-wuppertal.de/~xsc/preprints/prep_08_2.pdf",
keywords = "C-XSC, reliable computing, 100-digit challenge, reliable linear system solver, high performance computing, large dense linear systems",
note = {see also \cite{Kolberg:2009:NSP}},
remark = {see also Preprint BUW-WRSWT 2008/2},
bibdate = today,
bibsource = "http://drops.dagstuhl.de/opus/volltexte/2008/1447;
http://liinwww.ira.uka.de/bibliography//Misc/Dagstuhl_Publishing.html;
http://dblp.uni-trier.de/db/conf/dagstuhl/P8021.html#KolbergKZ08",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@InProceedings{Luther:2008:ACN,
title = {08021 Abstracts Collection - Numerical Validation in Current Hardware Architectures},
author = "Wolfram Luther and Annie A. M. Cuyt and Walter Kr{\"a}mer and Peter W. Markstein",
booktitle = "Numerical Validation in Current Hardware Architectures, 6.1. - 11.1.2008",
editor = "Annie A. M. Cuyt and Walter Kr{\"a}mer and Wolfram Luther and Peter W. Markstein",
series = "Dagstuhl Seminar Proceedings",
publisher = "Internationales Begegnungs- und Forschungszentrum
f{\"u}r Informatik (IBFI), Schloss Dagstuhl, Germany",
year = "2008",
volume = "08021",
ISSN = "1862-4405",
URL = "http://drops.dagstuhl.de/opus/volltexte/2008/1478",
other-URL = "http://drops.dagstuhl.de/portals/08021/",
note = "see also \cite{Cuyt:2009:NVC}",
keywords = "Computer arithmetic, arbitrary precision, floating-point arithmetic standardization, language
support, reliable libraries,high-precision special",
bibdate = today,
bibsource = "http://dblp.uni-trier.de/db/conf/dagstuhl/P8021.html#LutherCKM08,
http://liinwww.ira.uka.de/bibliography//Misc/Dagstuhl_Publishing.html",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@article {Popova:2008:VPS,
author = "Evgenija D. Popova and Walter Kr{\"a}mer",
title = {Visualizing parametric solution sets},
JOURNAL = "BIT",
FJOURNAL = "BIT. Numerical Mathematics",
volume = "48",
number = "1",
pages = "95--115",
month = mar,
year = "2008",
CODEN = "BITTEL, NBITAB",
ISSN = "0006-3835 (print), 1572-9125 (electronic)",
issn-l = "0006-3835",
MRCLASS = "15A06; 65G99; 65S05; 68U05",
DOI = "10.1007/s10543-007-0159-3",
URL = "http://dx.doi.org/10.1007/s10543-007-0159-3",
doi-url = "http://dx.doi.org/10.1007/s10543-007-0159-3",
preprint-url = "http://www2.math.uni-wuppertal.de/~xsc/preprints/prep_06_10-2.pdf",
journal-url = "http://link.springer.com/journal/10543",
ABSTRACT = {We characterize the boundary $\partial\Sigma^p$ of the solution set $\Sigma^p$ of a parametric linear system
$A(p)x = b(p)$ where the elements of the $n\times n$ matrix and the right-hand side vector depend on a number of parameters $p$
varying within interval bounds. The characterization of $\partial\Sigma^p$ is by means of pieces of parametric
hypersurfaces, the latter represented by their coordinate functions depending on corresponding subsets of $n-1$ parameters.
The presented approach has a direct application for efficient visualization of parametric solution sets by utilizing some
plotting functions supported by {\em Mathematica} and Maple.},
keywordss = "interval uncertainties; linear systems with dependent data; parametric solution set; visualization",
remark = "Available also as Preprint BUW-WRSWT 2006/10, Bergische Universit{\"a}t, Wuppertal, 2006",
bibdate = today,
bibsource = "http://rd.springer.com/export-citation/article/10.1007/s10543-007-0159-3.bib;
http://www.math.utah.edu/pub/tex/bib/bit.bib",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@InProceedings{Zimmer:2008:FPD,
author = "Michael Zimmer and Walter Kr{\"a}mer",
title = {Fast (Parallel) Dense Linear Interval Systems Solvers
in {C}-{XSC} Using Error Free Transformations and {BLAS}},
booktitle = "Numerical Validation in Current Hardware Architectures, 6.1. - 11.1.2008",
editor = "Annie A. M. Cuyt and Walter Kr{\"a}mer and Wolfram Luther and Peter W. Markstein",
series = "Dagstuhl Seminar Proceedings",
publisher = "Internationales Begegnungs- und Forschungszentrum f{\"u}r Informatik (IBFI), Schloss Dagstuhl, Germany",
year = "2008",
volume = "08021",
ISSN = "1862-4405",
URL = "http://drops.dagstuhl.de/opus/volltexte/2008/1436/pdf/08021.ZimmerMichael.Paper.1436.pdf",
keywords = "Error-free transformations, K-fold accuracy, accurate dot product, C-XSC, high accuracy, dense
linear systems, verified computation",
note = {see also \cite{Kraemer:2009:FPD}},
bibdate = today,
bibsource = "http://drops.dagstuhl.de/opus/volltexte/2008/1436;
http://liinwww.ira.uka.de/bibliography//Misc/Dagstuhl_Publishing.html;
http://dblp.uni-trier.de/db/conf/dagstuhl/P8021.html#ZimmerK08",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@incollection{Blomquist:2009:MSC,
author = "Frithjof Blomquist and Werner Hofschuster and Walter Kr{\"a}mer",
title = {A Modified Staggered Correction Arithmetic with Enhanced Accuracy and Very Wide Exponent Range},
booktitle = "Numerical Validation in Current Hardware Architectures",
editor = "Annie A. M. Cuyt and Walter Kr{\"a}mer and Wolfram Luther and Peter W. Markstein",
series = "Lecture Notes in Computer Science",
publisher = "Springer",
volume = "5492",
pages = "41--67",
year = "2009",
CODEN = "LNCSD9",
ISSN = "0302-9743",
ISBN = "978-3-642-01590-8 (print), 978-3-642-01591-5 (online)",
DOI = "10.1007/978-3-642-01591-5_4",
doi-url = "http://dx.doi.org/10.1007/978-3-642-01591-5_4",
URL = "http://link.springer.com/chapter/10.1007/978-3-642-01591-5_4",
MRCLASS = "65G20, 65G30, 65Y99, 37M99, 30-04",
keywords = "C-XSC; complex interval functions; interval computation; multiple precision; reliable numerical
computations; staggered correction; wide exponent range",
abstract = {A so called staggered precision arithmetic is a special kind of a multiple precision arithmetic based on the underlying floating point data format (typically IEEE double format) and fast floating point operations as well as exact dot product computations. Due to floating point limitations it is not an arbitrary precision arithmetic. However, it typically allows computations using several hundred mantissa digits. A set of new modified staggered arithmetics for real and complex data as well as for real interval and complex interval data with very wide exponent range is presented. Some applications will show the increased accuracy of computed results compared to ordinary staggered interval computations. The very wide exponent range of the new arithmetic operations allows computations far beyond the IEEE data formats. The new modified staggered arithmetics would be extremely fast if an exact dot product was available in hardware (the fused accumulate and add instruction is only one step in this direction).This paper describes work in progress. Updates of the software as well as additional documentation may be downloaded from our web site \url{http://www.math.uni-wuppertal.de/~xsc}.},
note = "see also \cite{Blomquist:2008:MSC}",
remark = "Conference on Numerical Validation in Current Hardware Architectures",
remark-2 = "Includes algorithms for division, $\exp(x)$, $(1 +
x)^n$, $\log(x)$, $\log(1 + x)$, and $\sqrt{x}$.
Staggered arithmetic represents numbers with tuples
$(e, x_1, x_2, ..., x_n)$ where $e$ is either integer
or a floating-point whole number, the $x_k$ are
floating-point, and a number has the value $2^e \sum_{k
= 1}^n x_k$. For interval arithmetic, the last element
is a pair of lower and upper bounds.",
bibdate = today,
bibsource = "http://rd.springer.com/export-citation/chapter/10.1007/978-3-642-01591-5_4.bib;
http://www.math.utah.edu/pub/tex/bib/elefunt.bib;
http://www.math.utah.edu/pub/tex/bib/fparith.bib;
http://dblp.uni-trier.de/db/conf/dagstuhl/nv2008.html#BlomquistHK08a",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@BOOK{Cuyt:2009:NVC,
editor = "Annie Cuyt and Walter Kr{\"a}mer and Wolfram Luther and Peter Markstein",
title = {Numerical Validation in Current Hardware Architectures},
series = "Lecture Notes in Computer Science",
volume = "5492",
year = "2009",
ISBN = "978-3-642-01590-8 (Print) 978-3-642-01591-5 (Online)",
publisher = "Springer Berlin Heidelberg",
DOI = "10.1007/978-3-642-01591-5",
URL = "http://dx.doi.org/10.1007/978-3-642-01591-5",
doi-URL = "http://dx.doi.org/10.1007/978-3-642-01591-5",
note = "see \cite{Cuyt:2008:SNV}, \cite{Luther:2008:ACN}",
remark = "International Dagstuhl Seminar, Dagstuhl Castle, Germany, January 6-11, 2008. Revised Papers.",
bibdate = today,
bibsource = "",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@incollection{Hofschuster:2009:CCR,
title = {{C}-{XSC} and Closely Related Software Packages},
author = "Werner Hofschuster and Walter Kr{\"a}mer and Markus Neher",
booktitle = "Numerical Validation in Current Hardware Architectures",
editor = "Annie A. M. Cuyt and Walter Kr{\"a}mer and Wolfram Luther and Peter W. Markstein",
series = "Lecture Notes in Computer Science",
publisher = "Springer",
year = "2009",
volume = "5492",
pages = "68--102",
CODEN = "LNCSD9",
ISSN = "0302-9743",
ISBN = "978-3-642-01590-8 (print), 978-3-642-01591-5 (online)",
DOI = "10.1007/978-3-642-01591-5_5",
URL = "http://dx.doi.org/10.1007/978-3-642-01591-5_5",
doi-URL = "http://dx.doi.org/10.1007/978-3-642-01591-5_5",
note = "see also \cite{Hofschuster:2008:CCR}",
remark = "Conference on Numerical Validation in Current Hardware Architectures",
MRCLASS = "68N30, 68N19, 65F99, 65G20, 65G30",
keywords = {C-XSC; mathematical software; reliable computing},
abstract = {C-XSC is an extensive and sophisticated C++ class library for the development and implementation of self-validating numerical software. Many numerical data types of distinct precision as well as operators and functions for these data types are provided by the library. Moreover, a large number of self-verifying numerical routines are integrated, and many additional packages for the reliable solution of numerical problems have been built on the C-XSC library. An MPI extension for C-XSC data types is available, enabling the efficient implementation of C-XSC software on parallel computers. In this paper, we present the basic features of C-XSC and we show by code examples that the development of sophisticated mathematical software delivering verified numerical results is considerably simplified when using C-XSC. Some features concerning complex interval arithmetic and complex interval functions (C-XSC, CoStLy, ACETAF) are discussed in more detail. All sample codes listed in this paper are available on the web page \url{http://www.math.uni-wuppertal.de/~xsc/cxsc/examples}.},
bibdate = today,
bibsource = "http://rd.springer.com/export-citation/chapter/10.1007/978-3-642-01591-5_5.bib;
http://dblp.uni-trier.de/db/conf/dagstuhl/nv2008.html#HofschusterKN08a",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@incollection{Kolberg:2009:NSP,
title = {A Note on Solving Problem 7 of the {SIAM} 100-Digit Challenge Using {C}-{XSC}},
author = "Mariana Luderitz Kolberg and Walter Kr{\"a}mer and Michael Zimmer",
booktitle = "Numerical Validation in Current Hardware Architectures",
editor = "Annie A. M. Cuyt and Walter Kr{\"a}mer and Wolfram Luther and Peter W. Markstein",
series = "Lecture Notes in Computer Science",
publisher = "Springer",
year = "2009",
volume = "5492",
pages = "250--261",
CODEN = "LNCSD9",
ISSN = "0302-9743",
ISBN = "978-3-642-01590-8 (print), 978-3-642-01591-5 (online)",
DOI = "10.1007/978-3-642-01591-5_16",
URL = "http://dx.doi.org/10.1007/978-3-642-01591-5_16",
doi-URL = "http://dx.doi.org/10.1007/978-3-642-01591-5_16",
preprint-URL = "http://www2.math.uni-wuppertal.de/~xsc/preprints/prep_08_2.pdf",
note = "see also \cite{Kolberg:2008:NSP}",
remark = "Conference on Numerical Validation in Current Hardware Architectures,
see also Preprint BUW-WRSWT 2008/2, Bergische Universit{\"a}t, Wuppertal, 2008.",
keywords = {100-digit challenge; C-XSC; high performance computing; large dense linear systems; reliable computing; reliable linear system solver},
abstract = {This paper presents the usage of a reliable parallel linear system solver to compute the solution of problem 7 of the SIAM 100-digitchallenge. Tests were executed on two different clusters: ALICEnext inWuppertal and XC1 in Karlsruhe. An approach to find all the 100 digits of the exact solution with maximum accuracy is also discussed and tested.},
bibdate = today,
bibsource = "http://rd.springer.com/export-citation/chapter/10.1007/978-3-642-01591-5_16.bib;
http://dblp.uni-trier.de/db/conf/dagstuhl/nv2008.html#KolbergKZ08a",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@incollection{Kraemer:2009:FPD,
author = "Walter Kr{\"a}mer and Michael Zimmer",
title = {Fast (Parallel) Dense Linear System Solvers in {C}-{XSC} Using Error Free Transformations and {BLAS}},
booktitle = "Numerical Validation in Current Hardware Architectures",
editor = "Annie A. M. Cuyt and Walter Kr{\"a}mer and Wolfram Luther and Peter W. Markstein",
series = "Lecture Notes in Computer Science",
publisher = "Springer",
year = "2009",
volume = "5492",
pages = "230--249",
CODEN = "LNCSD9",
ISSN = "0302-9743",
ISBN = "978-3-642-01590-8 (print), 978-3-642-01591-5 (online)",
DOI = "10.1007/978-3-642-01591-5_15",
URL = "http://dx.doi.org/10.1007/978-3-642-01591-5_15",
doi-URL = "http://dx.doi.org/10.1007/978-3-642-01591-5_15",
note = "see also \cite{Zimmer:2008:FPD}",
remark = "Conference on Numerical Validation in Current Hardware Architectures",
MRCLASS = "65H10, 15-04, 65G99, 65G10, 65-04, 68W15",
keywords = "selfverifying methods, large linear interval systems, DotK methods, parallelization, block cyclic distribution,
C-XSC, interval computations",
abstract = "Existing selfverifying solvers for dense linear (interval-) systems in C-XSC provide high accuracy, but are rather slow. A new set of solvers is presented, which are a lot faster than the existing solvers, without losing too much accuracy. This is achieved through two main changes. First, an alternative method for the computation of exact dot products based on the DotK-Algorithm is implemented. Then, optimized BLAS and LAPACK routines are used for the most costly parts, in terms of runtime, of the algorithm. Verified results are achieved by manipulating the rounding mode of the processor. Finally, an efficient parallel version of these solvers for distributed memory systems, based on ScaLAPACK, is presented, which allows to solve very large dense systems.
The new solver is compared to other solvers with respect to runtime and to numerical quality of the final result.",
bibdate = today,
bibsource = "http://rd.springer.com/export-citation/chapter/10.1007/978-3-642-01591-5_15.bib;
http://dblp.uni-trier.de/db/conf/dagstuhl/nv2008.html#KramerZ08",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@ARTICLE{Kraemer:2010:CAP,
AUTHOR = {Kr{\"a}mer, Walter},
TITLE = {Computer-assisted proofs and symbolic computations},
JOURNAL = {Serdica Journal of Computing},
VOLUME = {4},
YEAR = {2010},
NUMBER = {1},
PAGES = {73--84},
ISSN = {1312-6555},
URL = {http://serdica-comp.math.bas.bg/index.php/serdicajcomputing/article/view/91/94},
preprint-url = {http://www2.math.uni-wuppertal.de/~xsc/preprints/prep_09_3.pdf},
MRCLASS = {68T15, 65G20, 68W30},
ABSTRACT = {We discuss some main points of computer-assisted proofs based on reliable numerical computations. Such so-called self-validating numerical methods in combination with exact symbolic manipulations result in very powerful mathematical software tools. These tools allow proving mathematical statements (existence of a fixed point, of a solution of an ODE, of a zero of a continuous function, of a global minimum within a given range, etc.) using a digital computer. To validate the assertions of the underlying theorems fast finite precision arithmetic is used. The results are absolutely rigorous. To demonstrate the power of reliable symbolic-numeric computations we investigate in some details the verification of very long periodic orbits of chaotic dynamical systems. The verification is done directly in Maple, e.g. using the Maple Power Tool intpakX or, more efficiently, using the C++ class library C-XSC.},
bibdate = today,
bibsource = "http://www.ams.org/mathscinet/, MRNUMBER=2722265",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@INCOLLECTION{Kraemer:2010:HPV,
author = {Kr{\"a}mer, Walter},
title = {High Performance Verified Computing Using {C}-{XSC}},
booktitle = {IntMath-TSD: Interval Mathematics and Connections in Teaching and Scientific Development},
editor = {Reiser, R. H. S. and Pilla, M. L.},
publisher = {Universidade Federal de Pelotas, Ed. Universitria},
pages = {3--14},
year = {2010},
bibdate = today,
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@article {Kraemer:2010:VMS,
AUTHOR = {Kr{\"a}mer, Walter},
TITLE = {Verification methods and symbolic computations},
JOURNAL = {Albanian J. Math.},
FJOURNAL = {Albanian Journal of Mathematics},
publisher = {AulonaPress},
VOLUME = {4},
YEAR = {2010},
NUMBER = {4},
PAGES = {123--133},
ISSN = {1930-1235},
URL = {http://journals.aulonapress.com/index.php/ajm/article/view/268/338},
other-URL = {http://www.doaj.org/doaj?func=openurl\&genre=article\&issn=19301235\&date=2010\&volume=4\&issue=4\&spage=123},
MRCLASS = "68W30, 65Y15",
keywords = "computer-assisted proofs; self-verifying methods; arbitrary precision; interval functions; intpakX, C-XSC",
abstract = "Our intpakX package extends the computer algebra
system Maple. It allows, e.g., veried numerical
calculations (computer-assisted proofs) builton
arbitrary precision interval operations. Up to now,
only the basic operationsare supported in a guaranteed
way. Concerning higher mathematicalfunctions supported
in Maple, there are no data about their accuracies
available/published. Thus, it is not possible or at
least very hard to build arbitraryprecision interval
functions using Maple's intrinsic mathematical
functions(nevertheless, intpakX oers such function
implementations using someguard digits in an
experimental way, which - of course - is not really a
reliablemathematical approach).On the other hand there
are software packages supporting reliable
multipleprecision interval functions like C-XSC, the
MPFR and the MPFI libraries,and others. In the talk we
discuss the features of some of these libraries
indetail. We emphasize the dierent approaches
(arbitrary precision arithmetic,staggered correction
arithmetic, functions only for real arguments,
functionsfor complex arguments, ...) and the most
important resulting properties of thecorresponding
implementations. We also compare their performance and
wecomment on the actual integration of several of these
libraries in C-XSC. Themissing step is to bring
together C-XSC and computer algebra packages likeMaple
and Mathematica. Combining fast verication methods and
symboliccomputations deeply extends the range of
applications of rigorous mathematicalmethods.",
bibdate = today,
bibsource = "OAI-PMH server at www.doaj.org;
http://liinwww.ira.uka.de/bibliography//Misc/articles.doaj.org.html;
http://www.ams.org/mathscinet/, MRNUMBER=2755391",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@INCOLLECTION {Popova:2010:CFE,
AUTHOR = {Popova, Evgenija D. and Kr{\"a}mer, Walter},
TITLE = {Communicating functional expressions from {\em \uppercase{m}athematica} to {C-XSC}},
SERIES = {Lecture Notes in Computer Science},
BOOKTITLE = {Mathematical Software --- ICMS 2010},
EDITOR = {Fukuda, Komei and Hoeven, Joris van der and Joswig, Michael and Takayama, Nobuki},
PUBLISHER = {Springer},
ADDRESS = {Berlin Heidelberg},
volume = "6327",
YEAR = {2010},
CODEN = "LNCSD9",
PAGES = {354-365},
ISSN = "0302-9743 (print), 1611-3349 (electronic)",
issn-l = "0302-9743",
ISBN = {978-3-642-15581-9},
DOI = {10.1007/978-3-642-15582-6_56},
URL = {http://dx.doi.org/10.1007/978-3-642-15582-6_56},
doi-URL = {http://dx.doi.org/10.1007/978-3-642-15582-6_56},
book-url = "http://www.springerlink.com/content/978-3-642-15582-6",
KEYWORDS = {interoperability; functional expressions; C-XSC; {\em Mathematica}; MathLink},
ABSTRACT = {This work focuses on a mechanism (and software) which communicates (via MathLink protocol) and provides compatibility between the representation of nonlinear functions specified as Mathematica expressions and objects of suitable classes supported by the C-XSC automatic differentiation modules. The application of the developed communication software is demonstrated by MathLink compatible programs embedding in Mathematica the C-XSC modules for automatic differentiation as packages. The design methodology, some implementation issues and the use of the developed software are discussed.},
bibdate = today,
bibsource = "http://rd.springer.com/export-citation/chapter/10.1007/978-3-642-15582-6_56.bib;
http://www.math.utah.edu/pub/tex/bib/lncs2010a.bib;
http://www.math.utah.edu/pub/tex/bib/mathematica.bib;
http://dblp.uni-trier.de/db/conf/icms/icms2010.html#PopovaK10",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@TechReport{Popova:2010:ICA,
author = "Popova, Evgenija D. and Kr{\"a}mer, Walter and Russev, Milan",
title = {Integration of {C}-{XSC} Automatic Differentiation in {\uppercase{M}athematica}},
type = "Preprint",
number = "BUW-WRSWT 2010/1",
year = "2010",
institution = "Bergische Universit{\"a}t Wuppertal",
address = "Wuppertal, Germany",
url = "http://www2.math.uni-wuppertal.de/~xsc/preprints/prep_10_1.pdf",
language = {English},
remark = "Also published as Preprint 3/2010, IMI-BAS, Sofia, 2010,
\url{http://www.math.bas.bg/~epopova/papers/10-preprintAD.pdf}",
keywords = "",
abstract = "",
bibdate = today,
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@ARTICLE{Popova:2010:SCP,
AUTHOR = {Popova, Evgenija D. and Kolev, Lubomir and Kr{\"a}mer, Walter},
TITLE = {A Solver for Complex-Valued Parametric Linear Systems},
JOURNAL = {Serdica Journal of Computing},
VOLUME = {4},
YEAR = {2010},
NUMBER = {1},
PAGES = {123--132},
ISSN = {1312-6555},
URL = {http://serdica-comp.math.bas.bg/index.php/serdicajcomputing/article/download/96/99},
KEYWORDS = {linear systems, complex-valued interval parameters, electrical circuits},
ABSTRACT = {This work reports on a new software for solving linear systems involving affine-linear dependencies between complex-valued interval parameters. We discuss the implementation of a parametric residual iteration for linear interval systems by advanced communication between the system Mathematica and the library C-XSC supporting rigorous complex interval arithmetic. An example of AC electrical circuit illustrates the use of the presented software.},
bibdate = today,
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@ARTICLE{Zimmer:2010:ECL,
AUTHOR = {Zimmer, M. and Kr{\"a}mer, Walter and Bohlender, G. and Hofschuster, W.},
TITLE = {Extension of the {C}-{XSC} Library with Scalar Products with Selectable Accuracy},
JOURNAL = {Serdica Journal of Computing},
VOLUME = {4},
YEAR = {2010},
NUMBER = {3},
PAGES = {349--370},
ISSN = {1312-6555},
URL = {http://serdica-comp.math.bas.bg/index.php/serdicajcomputing/article/view/109/112},
preprint-url = {http://www2.math.uni-wuppertal.de/~xsc/preprints/prep_09_4.pdf},
ABSTRACT = {The C++ class library C-XSC for scientific computing has been extended with the possibility to compute scalar products with selectable accuracy in version 2.3.0. In previous versions, scalar products have always been computed exactly with the help of the so-called long accumulator. Additionally, optimized floating point computation of matrix and vector operations using BLAS-routines are added in C-XSC version 2.4.0. In this article the algorithms used and their implementations, as well as some potential pitfalls in the compilation, are described in more detail. Additionally, the theoretical background of the employed DotK algorithm and the necessary modifications of the concrete implementation in C-XSC are briefly explained. Run-time tests and numerical examples are presented as well.},
bibdate = today,
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@article {Zimmer:2010:SMV,
AUTHOR = {Zimmer, Michael and Kr{\"a}mer, Walter and Hofschuster, Werner},
TITLE = {Sparse matrices and vectors in {C}-{XSC}},
JOURNAL = {Reliab. Comput.},
FJOURNAL = {Reliable Computing},
VOLUME = {14},
number = {1},
YEAR = {2010},
PAGES = {138--160},
ISSN = {1573-1340},
MRCLASS = {65F50 (65-04 65F05)},
URL = {http://interval.louisiana.edu/reliable-computing-journal/volume-14/reliable-computing-14-pp-138-160.pdf},
preprint-url = {http://www2.math.uni-wuppertal.de/~xsc/preprints/prep_09_7.pdf},
keywords = {C-XSC; Compressed column storage; K-fold precision; Sparse data types; Sparse matrix algorithms},
abstract = {C-XSC is a C++ library for reliable scientific computing, which provides data types for dense vectors and matrices with real, complex, real interval and complex interval entries. These data types are easy to use and provide many helpful functionalities such as the ability to work with submatrices and subvectors. However, when dealing with sparse vectors, and especially with sparse matrices, these data types are inefficient. C-XSC version 2.4.0 added special types for sparse vectors and matrices that take advantage of the sparsity, both for performance and for memory consumption. This paper explains the data structures for and the implementation of these new types. Many examples and some performance tests with sparse matrices from real world applications are included.},
bibdate = today,
bibsource = "Zimmer:2010:SMV in http://liinwww.ira.uka.de/bibliography/Misc/RC.html;
http://www.ams.org/mathscinet/, MRNUMBER = 2811963",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@TechReport{Blomquist:2011:CLR,
author = "Blomquist, Frithjof and Hofschuster, Werner and Kr{\"a}mer, Walter",
title = {C-{XSC}-{L}angzahlarithmetiken f{\"u}r reelle und komplexe
{I}ntervalle basierend auf den {B}ibliotheken {MPFR} und {MPFI}},
type = "Preprint",
number = "BUW-WRSWT 2011/1",
year = "2011",
institution = "Bergische Universit{\"a}t Wuppertal",
address = "Wuppertal, Germany",
url = "http://www2.math.uni-wuppertal.de/~xsc/preprints/prep_11_1.pdf",
language = {German},
keywords = "",
abstract = "",
bibdate = today,
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@article{Kolberg:2011:EPS,
AUTHOR = {Kolberg, Mariana and Kr{\"a}mer, Walter and Zimmer, Michael},
TITLE = {Efficient parallel solvers for large dense systems of linear interval equations},
JOURNAL = {Reliab. Comput.},
FJOURNAL = {Reliable Computing},
VOLUME = {15},
YEAR = {2011},
NUMBER = {3},
PAGES = {193--206},
ISSN = {1573-1340},
URL = {http://interval.louisiana.edu/reliable-computing-journal/volume-15/no-3/reliable-computing-15-pp-193-206.pdf},
MRCLASS = {15A06, 65G30, 65Y05, 68W15},
keywords = {block cyclic distribution; C-XSC; DotK methods; interval computations; large linear interval systems; parallelization; self-verifying methods},
abstract = {Verified solvers for dense linear (interval-)systems require a lot of resources, both in terms of computing power and memory usage. Computing a verified solution of large dense linear systems (dimension $n > 10000$) on a single machine quickly approaches the limits of today's hardware. Therefore, an efficient parallel verified solver for distributed memory systems is needed. In this work we present such a solver, implemented in C++ and using the C-XSC library for scientific computing [10, 8]. The solver utilizes MPI [27] for communication between the nodes in the parallel environment and, where applicable, high performance ScaLAPACK [4] and BLAS [3] routines for fast computing times. High precision dot products [5, 18, 19, 21] are used to compute narrow enclosures of the solution of the system. We present test results on several high performance distributed memory systems with different architectures, which show that our solver achieves good results, both in terms of numerical accuracy as well as computing time, and is highly portable. Furthermore, even very large systems ($n > 100000$) can be solved given a cluster with sufficient resources.},
bibdate = today,
bibsource = "http://liinwww.ira.uka.de/bibliography/Misc/RC.html;
http://www.ams.org/mathscinet/, MRNUMBER=2819084 (2012f:15004)",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@incollection{Kraemer:2011:CPE,
author = {Kr{\"a}mer, Walter},
title = {C-{XSC}: {A} powerful environment for reliable computations in the natural and engineering sciences},
booktitle = {Biomedical Engineering and Informatics (BMEI), 2011 4th International Conference on},
editor = {{Yongsheng Ding et al.}},
year = {2011},
month= oct,
volume = {4},
pages = {2130--2134},
art_number = {6098697},
DOI = {10.1109/BMEI.2011.6098697},
url = {http://dx.doi.org/10.1109/BMEI.2011.6098697},
doi-url = {http://dx.doi.org/10.1109/BMEI.2011.6098697},
isbn = {978-1-4244-9351-7},
publisher = {IEEE},
remark = {Shanghai, China, 2011},
keywords = {C++ language; mathematics computing; software libraries; software packages; C++ class library C-XSC; arbitrary precision interval computations;logistic equation; numerical calculations; reliable computations; software packages; software tools; equations; libraries; logistics; mathematical model; numerical models; reliability; vectors},
abstract = {In recent years the C++ class library C-XSC (library for eXtended Scientific Computing) has been supplemented by several additional software packages. Most recently, a package allowing arbitrary precision interval computations has been added. In this note we present the most important features of C-XSC and its supplements. The tools allow very powerful self-validating numerical computations in all domains of engineering. Exemplarily we present some verified computations concerning the logistic equation, an often used, most simple, but nontrivial dynamical system. C-XSC establishes safety in numerical calculations, e.g. in solving linear or nonlinear equations. It also allows to handle uncertainties (errors in measurements, rounding errors, ...) in a guaranteed way. The software tools described in this note are all publicly available. Their source codes can be downloaded from our website \url{http://www2.math.uni-wuppertal.de/wrswt/xsc/cxsc_new.html}.},
bibdate = today,
bibsource = "http://ieeexplore.ieee.org/",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@inproceedings{Kraemer:2011:CSE,
author = {Kr{\"a}mer, Walter},
title = {C-{XSC}, a Sophisticated Environment for Reliable Computing},
booktitle = {Proceedings of the 4-th international conference on
Mathematical Aspects of Computer and Information Sciences (MACIS 2011)},
editor = {Ratschan, Stefan},
remark = {Beijing, China, October 19--21, 2011},
pages = {115--125},
year = {2011},
bibdate = today,
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@article{Popova:2011:CSS,
AUTHOR = {Popova, Evgenija D. and Kr{\"a}mer, Walter},
TITLE = {Characterization of {$AE$} solution sets to a class of parametric linear systems},
JOURNAL = {C. R. Acad. Bulgare Sci.},
FJOURNAL = {Comptes Rendus de l'Acad\'emie Bulgare des Sciences},
VOLUME = {64},
YEAR = {2011},
NUMBER = {3},
PAGES = {325--332},
ISSN = {1310-1331},
URL = {http://www.proceedings.bas.bg/PDF1/B_03-03.pdf},
MRCLASS = {65F05},
KEYWORDS = {linear systems; quantified interval parameters; solution set},
ABSTRACT = {Consider linear systems whose input data are affine-linear functions of uncertain parameters varying within given intervals. We discuss the so-called AE solution sets of such systems, where the parameters are quantified and all universally quantified parameters precede all existentially quantified ones. Some explicit descriptions of the AE parametric solution sets are derived in the special case of parametric systems where each parameter is involved in only one equation of the system (it may occur several times within this equation). For the general AE parametric solution sets these explicit characterizations lead to some necessary conditions for nonemptiness of such solution sets. Numerical examples illustrate the theory.},
bibdate = today,
bibsource = "http://www.ams.org/mathscinet/, MRNUMBER=2848711 (2012i:65052)",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@article{Popova:2011:ECN,
AUTHOR = {Popova, Evgenija D. and Kr{\"a}mer, Walter},
TITLE = {Embedding {C-XSC} nonlinear solvers in {\em \uppercase{m}athematica}},
JOURNAL = {C. R. Acad. Bulgare Sci.},
FJOURNAL = {Comptes Rendus de l'Acad\'emie Bulgare des Sciences},
VOLUME = {64},
YEAR = {2011},
NUMBER = {1},
PAGES = {11--20},
ISSN = {1310-1331},
URL = {http://www.proceedings.bas.bg/PDF1/B_01-02.pdf},
MRCLASS = {65F05},
KEYWORDS = {C-XSC; interoperability; interval software; {\em Mathematica}; MathLink},
ABSTRACT = {This work presents the integration of C-XSC nonlinear problem-solving modules based on automatic differentiation into {\em Mathematica} via MathLink protocol.},
bibdate = today,
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@TechReport{Blomquist:2012:UCL,
author = "Blomquist, Frithjof and Hofschuster, Werner and Kr{\"a}mer, Walter",
title = {{Umfangreiche C-XSC-Langzahlpakete f{\"u}r beliebig genaue reelle und komplexe Intervallrechnung}},
type = "Preprint",
number = "BUW-WRSWT 2012/2",
year = "2012",
institution = "Bergische Universit{\"a}t Wuppertal",
address = "Wuppertal, Germany",
url = "http://www2.math.uni-wuppertal.de/~xsc/preprints/prep_12_2.pdf",
language = {German},
keywords = "",
abstract = "",
bibdate = today,
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@incollection{Kraemer:2012:APC,
author = {Kr{\"a}mer, Walter and Blomquist, Frithjof},
title = {Arbitrary precision complex interval computations in {C}-{XSC}},
booktitle = {Parallel Processing and Applied Mathematics},
series = {Lecture Notes in Computer Science},
publisher = {Springer},
EDITOR = {Roman Wyrzykowski and Jack Dongarra and Konrad Karczewski and Jerzy Wasniewski},
year = {2012},
volume = {7204},
pages = {457--466},
ISBN = {978-3-642-31499-5 (print), 978-3-642-31500-8 (online)},
DOI = {10.1007/978-3-642-31500-8_47},
url = {http://dx.doi.org/10.1007/978-3-642-31500-8_47},
doi-url = {http://dx.doi.org/10.1007/978-3-642-31500-8_47},
remark = {9th International Conference, PPAM 2011, Torun, Poland, September 11-14, 2011. Revised Selected Papers, Part II},
keywords = {arbitrary precision; C-XSC; complex interval functions; complex intervals; MPFI; MPFR; reliable computations},
abstract = {Based on the libraries MPFR and MPFI for arbitrary precision real and arbitrary precision real interval computations and corresponding interfaces to the C++ class library C-XSC, the new data type MpfciClass (multiple precision floating-point complex intervals) and corresponding operations/functions for arbitrary precision complex intervals have been implemented. Our new package allows to code mathematical expressions for the complex interval data type in their usual mathematical notation yielding easy to read and self-documenting source code. Meanwhile, more than 30 elementary mathematical functions have been realized. At any point of the program the user may change the precision setting of the computation. The maximum precision of complex interval variables is only restricted by memory limitations. Its exponent range is very large. To the knowledge of the authors there is no comparable package (with respect to the features provided) available worldwide. The new package is written in C++. It is freely available from http://www2.math.uni-wuppertal.de/org/WRST/xsc/ cxscsoftware.html},
bibdate = today,
bibsource = {http://rd.springer.com/export-citation/chapter/10.1007/978-3-642-31500-8_47.bib;
http://dblp.uni-trier.de/db/conf/ppam/ppam2011-2.html#KramerB11},
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@Article{Kraemer:2012:MAP,
author = "Walter Kr{\"a}mer",
title = {Multiple\slash arbitrary precision interval computations in {C}-{XSC}},
journal = "Computing",
FJOURNAL = "Computing",
volume = "94",
number = "2--4",
pages = "229--241",
month = mar,
year = "2012",
CODEN = "CMPTA2",
ISSN = "0010-485X (print), 1436-5057 (electronic)",
issn-l = "0010-485X",
DOI = "10.1007/s00607-011-0174-8",
URL = "http://link.springer.com/article/10.1007/s00607-011-0174-8",
doi-url = "http://dx.doi.org/10.1007/s00607-011-0174-8",
journal-url = "http://link.springer.com/journal/607",
remark = "Special Issue on GAMM-IMACS International Symposium on
Scientific Computing, Computer Arithmetic and Validated Numerics (SCAN2010).",
MRCLASS = "65G40, 65Y99, 97N80",
keywords = "arbitrary precision; C-XSC; interval computations; interval Newton method; MPFI; MPFR; multiple precision; reliable computations",
abstract = "As a new feature, C-XSC provides so-called wrapper classes to some external arbitrary precision real and interval packages. Operator and function name overloading is used to give the user easy access to the arithmetic operations and mathematical functions provided by the underlying Ansi C packages. We will discuss briefly so-called staggered precision arithmetics based on exact scalar products. Such an arithmetic is available in C-XSC e.g. for multiple precision complex intervals. We also discuss the usage of the arbitrary precision arithmetic packages MPFR and MPFI, which are now accessible conveniently from within C-XSC via class interfaces. As a typical application, we will present an arbitrary precision interval Newton method to find the root(s) of a continuously differentiable function in a prescribed domain. The user only has to supply the expression for the function in the usual mathematical notation. The derivative needed in the interval Newton operator is computed using automatic differentiation based on the arbitrary precision interval operations. To demonstrate the power of the package we compute an enclosure of the zero of a model problem with guaranteed accuracy of more than 10 million decimal digits.",
bibdate = today,
bibsource = "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=0010-485X&volume=94&issue=2;
http://www.ams.org/mathscinet/, MRNUMBER=2891892;
http://www.math.utah.edu/pub/tex/bib/computing.bib;
http://www.math.utah.edu/pub/tex/bib/fparith.bib;
http://dblp.uni-trier.de/db/journals/computing/computing94.html#Kramer1",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@incollection{Kraemer:2012:UCH,
author = {Kr{\"a}mer, Walter and Zimmer, Michael and Hofschuster, W.},
title = {Using {C}-{XSC} for high performance verified computing},
booktitle = {Applied Parallel and Scientific Computing},
series = {Lecture Notes in Computer Science},
editor = {J{\'o}nasson, Kristj{\'a}n},
ISBN = "978-3-642-28144-0 (print), 978-3-642-28145-7 (online)",
publisher = {Springer},
year = {2012},
volume = {7134},
pages = {168--178},
DOI = {10.1007/978-3-642-28145-7_17},
url = {http://dx.doi.org/10.1007/978-3-642-28145-7_17},
doi-url = {http://dx.doi.org/10.1007/978-3-642-28145-7_17},
remark = {10th International Conference, PARA 2010, Reykjav\'ik, Iceland, June 6-9, 2010, Revised Selected Papers, Part II},
keywords = {BLAS; C-XSC; compiler optimizations; dot product computation; error free transformation;
high performance computing; MPI; openMP},
abstract = {C-XSC is a C++ class library for scientific computing, with its main focus on reliable
interval computations. Recently, several changes and new features have been implemented, making C-XSC
much more suitable for tasks in high performance computing. However, these changes require that users take
several factors into consideration when writing and compiling programs using C-XSC to get the best possible performance while still maintaining a sufficient level of numerical accuracy. This paper gives an overview of the most important points concerning these factors and tries to give background information and recommendations to the end user for the implementation of efficient C-XSC programs.},
bibdate = today,
bibsource = "http://rd.springer.com/export-citation/chapter/10.1007/978-3-642-28145-7_17.bib;
http://dblp.uni-trier.de/db/conf/para/para2010-2.html#KramerZH10",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@Article{Zimmer:2012:SVS,
author = "Michael Zimmer and Walter Kr{\"a}mer and Evgenija D. Popova",
title = {Solvers for the verified solution of parametric linear systems},
journal = "Computing",
FJOURNAL = "Computing",
volume = "94",
number = "2--4",
pages = "109--123",
month = mar,
year = "2012",
CODEN = "CMPTA2",
ISSN = "0010-485X (print), 1436-5057 (electronic)",
issn-l = "0010-485X",
DOI = "10.1007/s00607-011-0170-z",
url = "http://dx.doi.org/10.1007/s00607-011-0170-z",
doi-url = "http://dx.doi.org/10.1007/s00607-011-0170-z",
journal-url = "http://link.springer.com/journal/607",
remark = "Special Issue on GAMM-IMACS International Symposium on
Scientific Computing, Computer Arithmetic and Validated Numerics (SCAN2010).",
MRCLASS = "65F05; 65F50; 65G20; 65G99",
keywords = {C-XSC; parameter dependent system; self-verifying solver; sparse data structure},
ABSTRACT = {We present a newly developed version of our solvers for the verified solution of dense parametric linear systems, i.e. linear systems whose system matrix and right-hand side depend affine-linearly on parameters that vary inside prescribed intervals. The solvers use our C++ class library for reliable computing, C-XSC. The C-XSC library provides many features, especially easy to handle data types for dense and sparse matrices and vectors and the ability to compute dot products and dot product expressions in arbitrary precision. The new solvers can use either sparse or dense matrices as the coefficient matrices for the parameters. The use of sparse coefficient matrices can result in huge improvements in both performance and memory consumption. BLAS and LAPACK routines are used where applicable, and OpenMP is used for the parallelization on multi-core and multi-processor systems. The solvers also provide the ability to compute not only an outer but also a componentwise inner enclosure of the solution set of the system and to choose between two versions of the algorithm, one being very fast and one giving sharp results and extending the range of solvable systems. We give some examples for parametric linear systems (also from real world examples such as worst-case tolerance analysis of linear electric circuits), give performance measurements of our solvers and also demonstrate that they scale very well when using multiple cores or processors.},
bibdate = today,
bibsource = "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=0010-485X&volume=94&issue=2;
http://www.ams.org/mathscinet/, MRNUMBER=2891884;
http://dblp.uni-trier.de/db/journals/computing/computing94.html#ZimmerKP12;
http://www.math.utah.edu/pub/tex/bib/computing.bib",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@article{Kraemer:2013:HPV,
AUTHOR = {Kr{\"a}mer, Walter},
TITLE = {High performance verified computing using {C}-{XSC}},
JOURNAL = {Comput. Appl. Math.},
FJOURNAL = {Computational and Applied Mathematics},
VOLUME = {32},
YEAR = {2013},
NUMBER = {3},
PAGES = {385--400},
ISSN = {0101-8205},
DOI = {10.1007/s40314-013-0028-4},
URL = {http://dx.doi.org/10.1007/s40314-013-0028-4},
doi-URL = {http://dx.doi.org/10.1007/s40314-013-0028-4},
KEYWORDS = {verified computing, self-validating methods, high performance computing, parallelization, thread-safety, sparse methods, C-XSC, 65G20, 65G30},
ABSTRACT = {So called self-validating or self-verifying numerical methods allow to prove mathematical statements (existence of a fixed point, of a solution of an ODE, of a zero of a continuous function, of a global minimum within a given range, etc.) using a digital computer. To validate the assertions of the underlying mathematical theorems only fast finite precision machine arithmetic is used. The results are absolutely rigorous. We report on the accuracy as well as on the efficiency of the C++ class library C-XSC, our well known open source software tool designed to facilitate self-verifying numerical calculations. We focus mainly on solvers for dense and sparse interval linear systems. In recent years, these solvers have been improved significantly with respect to high performance computing within our bilateral Probral project HPVC (see Acknowledgments). As a motivating nontrivial example, where we need in an intermediate step an efficient solver for large dense interval linear systems, the computation of a verified functional enclosure for the solution of an integral equation is briefly discussed. The newest version C-XSC 2.5.1 released on June 9, 2011 allows using C-XSC in multi-threaded environments. The library as well as some further packages not mentioned in this paper are open source and freely available from the web site of the author's research group Scientific Computing/Software Engineering at the University of Wuppertal:
\url{http://www2.math.uni-wuppertal.de/org/WRST/index_de.html}.},
bibdate = today,
bibsource = "http://www.ams.org/mathscinet/, MRNUMBER=3120127;
http://rd.springer.com/export-citation/article/10.1007/s40314-013-0028-4.bib",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
@Article{Zimmer:2013:OCT,
title = {An overview of {C}-{XSC} as a tool for interval arithmetic and its application in computing verified
uncertain probabilistic models under {D}empster-{S}hafer theory},
author = "Michael Zimmer and Gabor Rebner and Walter Kr{\"a}mer",
journal = "Soft Computing",
fjournal = {Soft Computing},
year = "2013",
number = "8",
volume = "17",
pages = "1453--1465",
DOI = "10.1007/s00500-013-1014-x",
URL = "http://dx.doi.org/10.1007/s00500-013-1014-x",
doi-URL = "http://dx.doi.org/10.1007/s00500-013-1014-x",
ISSN = {1432-7643 (print), 1433-7479 (online)},
keywords = {BLAS; C-XSC; DotK; DSI; INTLAB; uncertain data; verified results},
abstract = {Interval arithmetic can be a useful tool in soft computing. However, working with intervals requires specialized algorithms and appropriate data structures. In this paper, we give an overview of the C++ library C-XSC, which provides many useful data types and functions for (verified) scientific computing, with a special focus on interval arithmetic. We describe its basic features and focus especially on some recent new features that significantly broaden the range of uses of C-XSC, such as dot products in K-fold double precision, sparse data types and BLAS support. In a second section, we describe an application of C-XSC in soft computing in the form of an interface between C-XSC and MATLAB for the DSI-Toolbox, which combines Dempster-Shafer theory to model uncertain data with verified interval arithmetic. For some applications, utilizing C-XSC in MATLAB (or as a standalone) can be more effective than the widely used INTLAB extension for MATLAB, due to the lack of interpretation overhead. As an example, we use C-XSC for the normalization function in the DSI-Toolbox and compare the results and performance to those provided by INTLAB. Using C-XSC in MATLAB also extends the functionality of MATLAB/INTLAB. As an example, the interval error function of C-XSC (this function is not provided by INTLAB) is utilized via a MEX-interface for the verified sampling of the cumulative normal distribution.},
bibdate = today,
bibsource = "http://rd.springer.com/export-citation/article/10.1007/s00500-013-1014-x.bib;
http://dblp.uni-trier.de/db/journals/soco/soco17.html#ZimmerRK13",
acknowledgement = ack-edp,
author-dates = "1952--2014 (WK)",
}
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